Recent zbMATH articles in MSC 47https://www.zbmath.org/atom/cc/472022-05-16T20:40:13.078697ZWerkzeugLocation of Ritz values in the numerical range of normal matriceshttps://www.zbmath.org/1483.150052022-05-16T20:40:13.078697Z"Dela Rosa, Kennett L."https://www.zbmath.org/authors/?q=ai:dela-rosa.kennett-l"Woerdeman, Hugo J."https://www.zbmath.org/authors/?q=ai:woerdeman.hugo-jSummary: Let \(\mu_1\) be a complex number in the numerical range \(W(A)\) of a normal matrix \(A\). In the case when no eigenvalues of \(A\) lie in the interior of \(W(A)\), we identify the smallest convex region containing all possible complex numbers \(\mu_2\) for which \(\begin{bmatrix} \mu_1 & \ast \\ 0 & \mu_2 \end{bmatrix}\) is a 2-by-2 compression of \(A\).Sharp nonzero lower bounds for the Schur product theoremhttps://www.zbmath.org/1483.150102022-05-16T20:40:13.078697Z"Khare, Apoorva"https://www.zbmath.org/authors/?q=ai:khare.apoorvaTo sketch the background of this interesting paper, let us recall a few basic facts from matrix analysis and linear algebra. Fix \(m, n \in \mathbb{N}\). Let \(M_{m,n}(\mathbb{F}) \equiv \mathbb{F}^{m \times n}\) denote the set of all \(m \times n\) matrices with entries in \(\mathbb{F}\), where \(\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}\). A matrix \(A \in M_n(\mathbb{F}) \equiv M_{n,n}(\mathbb{F})\) is positive semidefinite (resp., positive definite) in \(\mathbb{M}_n(\mathbb{F})\) if and only if the following two conditions are satisfied:
\begin{itemize}
\item[(i)] \(A = A^\ast\) (i.e., \(A\) is Hermitian);
\item[(ii)] \(x^\ast A x \geq 0\) (respectively \(x^\ast A x > 0\)) for all \(x \in {\mathbb{F}^n}\setminus\{0\}\).
\end{itemize}
If \(A \in \mathbb{M}_n(\mathbb{R})\), then \(A^\ast = A^\top\). In the complex case, condition (i) is unnecessary. However, the inclusion of (i) guarantees that \(A \in M_n(\mathbb{R})\) is positive semidefinite in \(M_n(\mathbb{R})\) if and only if \(A \in M_n(\mathbb{R}) \subseteq M_n(\mathbb{C})\) is positive semidefinite in \(M_n(\mathbb{C})\). To recognise this fact, we just have to examine the example of the non-symmetric real \(2 \times 2\)-matrix \(\left(\begin{smallmatrix} 0 & 1\\ -1 & 0 \end{smallmatrix}\right)\).
Let \(\mathbb{P}_n(\mathbb{F}) \subseteq \mathbb{M}_n(\mathbb{F})\) denote the set of all positive semidefinite \(n \times n\) matrices with entries in \(\mathbb{F}\) (in the following simply abbreviated as PSD).
The considerably rich structure of the convex cone \(\mathbb{P}_n(\mathbb{F})\) is not only essential in linear algebra and matrix analysis itself. \(\mathbb{P}_n(\mathbb{F})\) also plays a key role in conic optimisation (in particular, semidefinite programming), quantum information theory, computational complexity and spectral graph theory.
If \(A, B \in \mathbb{P}_n(\mathbb{F})\), it is a well-known fact that in general the standard matrix product \(AB\) is not positive semidefinite. In fact, \(AB \in \mathbb{P}_n(\mathbb{F})\) if and only if \(AB\) is Hermitian, which is equivalent to \(AB = BA\), i.e., \(A\) and \(B\) commute (see [\textit{A. R. Meenakshi} and \textit{C. Rajian}, Linear Algebra Appl. 295, No. 1--3, 3--6 (1999; Zbl 0940.15022)]).
The situation changes completely when the Hadamard product of matrices is considered instead. If \(C = (c_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\) and \(D = (d_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\), the \textit{Hadamard product of \(C\) and \(D\)} is defined as
\[
C \circ D : = (c_{ij}\,d_{ij}) \quad ((i,j) \in [m] \times [n])\,.
\]
The Hadamard product is sometimes called the \textit{entrywise product} for obvious reasons, or the \textit{Schur product}, because of some early and basic results about the product obtained by Schur (see [\textit{R. A. Horn} and \textit{C. R. Johnson}, Topics in matrix analysis. Cambridge etc.: Cambridge University Press (1991; Zbl 0729.15001)]).
(NB: In my opinion, the symbolic notation \(\circ\) perhaps could lead to a minor ambiguity, since quite regularly, \(\circ\) denotes composition of mappings. In [\textit{V. Paulsen}, Completely bounded maps and operator algebras. Cambridge: Cambridge University Press (2002; Zbl 1029.47003)], the symbol \(\ast\) is used instead. Also \(\odot\) could be a useful substitution for \(\circ\).)
Like the usual matrix product, the distributive law also holds for the Hadamard product: \(A\circ(B + C) = A \circ B + A \circ C\). Unlike the usual matrix product, the Hadamard product is commutative: \(A \circ B = B \circ A\).
In the real subspace of Hermitian matrices, the most common order relation is the \textit{Loewner partial order}. It is induced by the cone \(\mathbb{P}_n(\mathbb{F})\). By definition, \(A \geq B\) if and only if both \(A, B\) are Hermitian and \(A-B \in \mathbb{P}_n(\mathbb{F})\). A seminal result by \textit{I. Schur} [J. Reine Angew. Math. 140, 1--28 (1911; JFM 42.0367.01)], nowadays known as ``Schur product theorem'', asserts that if \(A \geq 0\) and \(B \geq 0\), then also \(A \circ B \geq 0\) (see [Horn and Johnson, loc. cit.], Chapter 5.2). Moreover, if we implement the (unique) positive semidefinite root of a PSD matrix (\(A = A^{1/2}\,A^{1/2}\) for all \(A \in \mathbb{P}_n(\mathbb{F})\)) into the trace, we reobtain the well-known fact that \(\mathbb{P}_n(\mathbb{F})\) is a self-dual cone (for both fields), meaning that
\[
A \geq 0 \text{ if and only if } \langle A, B\rangle_F : = \operatorname{tr}(A B^\ast) = \operatorname{tr}(A B) \geq 0 \ \text{ for all } B \in \mathbb{P}_n(\mathbb{F}).
\]
Thereby,
\[
\mathbb{M}_{m,n}(\mathbb{F}) \times \mathbb{M}_{m,n}(\mathbb{F}) \ni (C, D) \mapsto \langle C, D\rangle_F : = \operatorname{tr}(C D^\ast) = \operatorname{tr}(D^\ast\,C)
\]
denotes the Frobenius inner product. Because of Hölder's inequality, the matrix \(C \in \mathbb{M}_{m,n}(\mathbb{F})\) can be identified as bounded linear operator from \(l_p^n\) to \(l_q^m\) (of finite rank) which satisfies \(\Vert A \Vert \leq \big(\sum_{i = 1}^m \sum_{j = 1}^n \vert a_{ij}\vert^q\big)^{1/q}\), for \(1 \leq p, q \leq \infty\) with \(\frac{1}{p} + \frac{1}{q} = 1\). Thus, if we view \(C\) as Hilbert-Schmidt operator from the Hilbert space \(l_2^n\) to the Hilbert space \(l_2^m\), the Frobenius inner product coincides with the Hilbert-Schmidt inner product of \(C \in \mathcal{S}_2(l_2^n, l_2^m)\).
The paper consists of three parts. In the first part, the main result (Theorem A) enhances a result of \textit{J. Vybíral}, developed in [Adv. Math. 368, Article ID 107140, 8 p. (2020; Zbl 1441.15024), Theorem 1]. The author improves Vybíral's \textit{positive} lower bound (with respect to the Loewner partial order, induced by PSD matrices). That lower bound appears in a stronger version of the Schur product theorem, also introduced by Vybíral. Moreover, Theorem A unveils that the emerging positive constant as part of the lower bound is the maximum possible one. It cannot be further increased.
Besides an application of the Cauchy-Schwarz inequality with respect to the Frobenius inner product, the trace equality (1.12) plays a significant role in the proof of Theorem A.
We would like to emphasize that (1.12) itself can be deduced from the following simple fact, which however reveals a further direct link between the Hadamard product and the standard matrix product:
\[
M \circ uv^\top = D_u\,M\,D_v \text{ for all } M \in \mathbb{M}_{m,n}(\mathbb{F}), (u,v) \in \mathbb{F}^m \times \mathbb{F}^n\,,
\]
where \(D_x\) denotes the \(p \times p\)-diagonal matrix, whose \((i,j)\)'th entry is given by \(\delta_{ij}\,x_j\), for any \(x = (x_1, \ldots, x_p)^\top \in \mathbb{F}^p\). Here \(\delta_{ij}\) is the Kronecker delta.
The first part concludes with a few (fairly technical) refinements of Theorem A including the determination of an upper bound and an in-depth discussion of special cases of Theorem A which were published in the past, primarily by Vybíral.
In the second part, the author develops a ``suitably modified'' version of Theorem A which produces a non-trivial positive lower bound in the case of Hilbert-Schmidt operators between arbitrary, not necessarily finite-dimensional Hilbert spaces (see Theorem 2.4).
In the third and last part a few applications of Theorem A, including the important entrywise calculus on classes of positive matrices are touched briefly. The latter also plays a crucial role regarding an approximation of the upper bound of the real and complex Grothendieck constant in the famous Grothendieck inequality; a fact, based on my own research activities, not covered here. Complex kernels with lower bounds are introduced, and related non-trivial PSD matrices are listed. Even indications for future research problems are sketched. In this respect, I would like to add a further research question: could Theorem A and its applications to the entrywise calculus on classes of positive matrices even become useful to improve the already existing \textit{lower} bounds of the complex and real Grothendieck constant?
Finally, as a minor and weak ``criticism'', let me point out that it would be very helpful for the reader to see explicitly on which field \(\mathbb{F} \in \{\mathbb{R},\mathbb{C}\}\) a statement about \(\mathbb{P}_n(\mathbb{F})\) is referred to, mainly to work out what results hold for both fields simultaneously. In my view, the primarily considered field is \(\mathbb{C}\).
Reviewer: Frank Oertel (London)Partial determinant inequalities for positive semidefinite block matriceshttps://www.zbmath.org/1483.150112022-05-16T20:40:13.078697Z"Li, Yongtao"https://www.zbmath.org/authors/?q=ai:li.yongtao"Lin, Xiqin"https://www.zbmath.org/authors/?q=ai:lin.xiqin"Feng, Lihua"https://www.zbmath.org/authors/?q=ai:feng.lihuaThere are many reasons to study block matrices \(A=[A_{ij}]\) with each block \(A_{ij}\) square. Let \(f\) be a functional on the space of square matrices. Applying \(f\) to each block of \(A\) results in a smaller (sized) matrix \([f(A_{ij})]\). Research around \([f(A_{ij})]\) was initiated by several mathematicians in the 60s of the last century. For specific \(f\), people now call \([\text{tr}(A_{ij})]\) and \([\det(A_{ij})]\) the partial trace and the partial determinant of \(A\), respectively. The authors consider a general situation when \(f\) is any generalized matrix function, named by them partial matrix function. Some known inequalities on determinants and partial determinants are extended. In the final part of the paper, an interesting conjecture is proposed.
Reviewer: Minghua Lin (Xi'an)A note on the boundary of the Birkhoff-James \(\epsilon\)-orthogonality setshttps://www.zbmath.org/1483.150122022-05-16T20:40:13.078697Z"Katsouleas, Georgios"https://www.zbmath.org/authors/?q=ai:katsouleas.georgios"Panagakou, Vasiliki"https://www.zbmath.org/authors/?q=ai:panagakou.vasiliki"Psarrakos, Panayiotis"https://www.zbmath.org/authors/?q=ai:psarrakos.panayiotis-jThe numerical range of a matrix polynomial \(P\) is a generalization of the numerical range (field of values) of a matrix \(A\). The classical case is obtained for \(P(z) = zI - A\). Recently, the notions of Birkhoff-James \(\varepsilon\)-orthogonality sets of rectangular matrices and elements of a complex normed linear space have been introduced as natural generalizations of the standard numerical range. In this paper, a characterization for the corners of the Birkhoff-James-\(\varepsilon\)-sets of vectors and vector-valued polynomials is obtained. A randomized algorithm for approximating their boundaries is also proposed.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Matricial radius: a relation of numerical radius with matricial rangehttps://www.zbmath.org/1483.150132022-05-16T20:40:13.078697Z"Kian, Mohsen"https://www.zbmath.org/authors/?q=ai:kian.mohsen"Dehghani, Mahdi"https://www.zbmath.org/authors/?q=ai:dehghani.mahdi"Sattari, Mostafa"https://www.zbmath.org/authors/?q=ai:sattari.mostafaThe matricial range of an operator (on a Hilbert space) is a matrix valued extension of the numerical range. It would be natural to define the matricial radius of an operator to be the maximum norm of the elements of its matricial range. This definition can be not very interesting, as the matricial radius defined in this way would be equal to the norm of the operator. This paper proposes two other candidates for a more meaningful notion of matricial radius. The authors find that neither of them is a ``proper'' extension of the numerical radius based on the matricial range and wonder whether such an extension exists at all.
Reviewer: Minghua Lin (Xi'an)Asymptotically optimal multi-pavinghttps://www.zbmath.org/1483.150142022-05-16T20:40:13.078697Z"Ravichandran, Mohan"https://www.zbmath.org/authors/?q=ai:ravichandran.mohan"Srivastava, Nikhil"https://www.zbmath.org/authors/?q=ai:srivastava.nikhilAs a consequence of the solution of the Kadison-Singer problem in [\textit{A. W. Marcus} et al., Ann. Math. (2) 182, No. 1, 327--350 (2015; Zbl 1332.46056)], it is known that every zero-diagonal matrix admits a nontrivial paving with dimension independent bounds.
In this nice paper, it is shown that given \(k\) zero-diagonal \(n\times n\) Hermitian matrices \(A_1, \dots, A_k\) and \(\varepsilon > 0\), there is a simultaneous paving, i.e., there are diagonal projections \(P_1,\dots , P_r\) with \(r \le 18k/{\varepsilon^2}\) and \(\sum_{j\le r} P_j = I\) such that \(\|P_jA_iP_j\| \le \varepsilon \|A_i\|\). Furthermore, it is proved that every square zero-diagonal complex matrix can be \(\varepsilon\)-paved using \(O(\varepsilon^{-2})\) blocks. This is also used to strengthen a result by \textit{W. B. Johnson} et al. [Proc. Natl. Acad. Sci. USA 110, No. 48, 19251--19255 (2013; Zbl 1297.15027)]
on commutator representations of zero trace matrices.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Cayley-Hamilton theorem in the min-plus algebrahttps://www.zbmath.org/1483.150222022-05-16T20:40:13.078697Z"Siswanto"https://www.zbmath.org/authors/?q=ai:siswanto.nurhadi"Nurhayati, N."https://www.zbmath.org/authors/?q=ai:nurhayati.nunung"Pangadi"https://www.zbmath.org/authors/?q=ai:pangadi.Summary: Max-plus algebra is the set of \(\mathbb{R}_\varepsilon = \mathbb{R} \cup\{\varepsilon \}\) where \(\mathbb{R}\) is a set of all real numbers and \(\varepsilon = -\infty\) which is equipped with max \((\oplus)\) and plus \((\otimes)\) operations. The structure of max-plus algebra is semi-field. Max-plus algebra has structural similarities to conventional algebra. Because of these similarities, concept in conventional algebra such as the Cayley-Hamilton theorem has max-algebraic equivalence. Another semi-field is min-plus algebra, so that the Cayley-Hamilton theorem also has a min-plus algebraic equivalence. In this paper, we will show how the Cayley-Hamilton theorem can be proved in the min-plus algebra.Jordan higher derivations of incidence algebrashttps://www.zbmath.org/1483.160462022-05-16T20:40:13.078697Z"Chen, Lizhen"https://www.zbmath.org/authors/?q=ai:chen.lizhen"Xiao, Zhankui"https://www.zbmath.org/authors/?q=ai:xiao.zhankuiSummary: Let \(\mathcal{R}\) be a 2-torsion-free commutative ring with unity and \(X\) be a locally finite pre-ordered set. We prove in this paper that every Jordan higher derivation on the incidence algebra \(I(X,\mathcal{R})\) is a higher derivation. By the way, we also provide a new proof of the known fact that every Jordan derivation of \(I(X,\mathcal{R})\) is a derivation.Rate of growth of distributionally chaotic functionshttps://www.zbmath.org/1483.300592022-05-16T20:40:13.078697Z"Gilmore, Clifford"https://www.zbmath.org/authors/?q=ai:gilmore.clifford"Martínez-Giménez, Félix"https://www.zbmath.org/authors/?q=ai:martinez-gimenez.felix"Peris, Alfred"https://www.zbmath.org/authors/?q=ai:peris.alfredoSummary: We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for \(D\)-distributionally chaotic entire functions, where growth is in terms of average \(L^p\)-norms on spheres of radius \(r>0\) as \(r\rightarrow\infty\), for \(1\leq p\leq\infty\). We compute growth estimates of \(\partial/\partial x_k\)-distributionally chaotic harmonic functions in terms of the average \(L^2\)-norm on spheres of radius \(r>0\) as \(r\rightarrow\infty\). We also calculate sup-norm growth estimates of distributionally chaotic harmonic functions in the case of the partial differentiation operators \(D^\alpha\).The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domainhttps://www.zbmath.org/1483.300712022-05-16T20:40:13.078697Z"Gençtürk, İlker"https://www.zbmath.org/authors/?q=ai:gencturk.ilkerSummary: In this study, by using some integral representations formulas, we study solvability conditions and explicit solution of the Dirichlet-Neumann problem, an example for a combined boundary value problem, for the Bitsadze equation in a ring domain.Noether property and approximate solution of the Riemann boundary value problem on closed curveshttps://www.zbmath.org/1483.300742022-05-16T20:40:13.078697Z"Bory-Reyes, Juan"https://www.zbmath.org/authors/?q=ai:bory-reyes.juan|moreno-garcia.tania"Katz, David"https://www.zbmath.org/authors/?q=ai:katz.david-f|katz.david-bBased on classic methods, the authors analyse the Noether property of a Riemann boundary value problem in the Banach algebra of continuous functions over closed curves, as well as consequent approximate solutions (by using quasi-Fredholm operators).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Composition of analytic paraproductshttps://www.zbmath.org/1483.300962022-05-16T20:40:13.078697Z"Aleman, Alexandru"https://www.zbmath.org/authors/?q=ai:aleman.alexandru"Cascante, Carme"https://www.zbmath.org/authors/?q=ai:cascante.carme"Fàbrega, Joan"https://www.zbmath.org/authors/?q=ai:fabrega.joan"Pascuas, Daniel"https://www.zbmath.org/authors/?q=ai:pascuas.daniel"Peláez, José Ángel"https://www.zbmath.org/authors/?q=ai:pelaez.jose-angelSummary: For a fixed analytic function \(g\) on the unit disc \(\mathbb{D}\), we consider the analytic paraproducts induced by \(g\), which are defined by \(T_g f(z)=\int_0^z f(\zeta)g^\prime(\zeta)d\zeta\), \(S_g f(z)=\int_0^z f^\prime(\zeta) g(\zeta)d\zeta\), and \(M_g f(z)=f(z) g(z)\). The boundedness of these operators on various spaces of analytic functions on \(\mathbb{D}\) is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example \(T_g^2\), \(T_gS_g\), \(M_g T_g\), etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol \(g\). In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol \(g\) than the case of a single paraproduct.A representation theoretic explanation of the Borcea-Brändén characterizationhttps://www.zbmath.org/1483.320022022-05-16T20:40:13.078697Z"Leake, Jonathan"https://www.zbmath.org/authors/?q=ai:leake.jonathan-dSummary: In [Invent. Math. 177, No. 3, 541--569 (2009; Zbl 1175.47032)], \textit{J. Borcea} and \textit{P. Brändén} characterized all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace's theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea-Brändén result to characterize a certain subclass of the linear operators which preserve such polynomials.Nonlinear Hadamard fractional boundary value problems with different ordershttps://www.zbmath.org/1483.340092022-05-16T20:40:13.078697Z"Abuasbeh, Kinda"https://www.zbmath.org/authors/?q=ai:abuasbeh.kinda"Awadalla, Muath"https://www.zbmath.org/authors/?q=ai:awadalla.muath-m"Jneid, Maher"https://www.zbmath.org/authors/?q=ai:jneid.maherSummary: In this paper, we study the existence (uniqueness) of solutions for nonlinear fractional differential equations with different orders of Hadamard fractional derivatives involving associated with nonlocal boundary conditions. Several fixed point theorems are used for sufficient conditions of the existence (uniqueness) of solutions to nonlinear differential equations such as Banach's contraction principle, the Leray-Schauder nonlinear alternative, and Krasnoselskii's fixed point theorem. Applications of the main results are also presented.On coupled systems of Lidstone-type boundary value problemshttps://www.zbmath.org/1483.340232022-05-16T20:40:13.078697Z"de Sousa, Robert"https://www.zbmath.org/authors/?q=ai:de-sousa.robert"Minhós, Feliz"https://www.zbmath.org/authors/?q=ai:minhos.feliz-manuel"Fialho, João"https://www.zbmath.org/authors/?q=ai:fialho.joao-fSummary: This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.Solvability of inclusions of Hammerstein typehttps://www.zbmath.org/1483.340342022-05-16T20:40:13.078697Z"Pietkun, Radosław"https://www.zbmath.org/authors/?q=ai:pietkun.radoslawSummary: We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces with the aid of some recently proven continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of \(m\)-accretive multivalued differential equation, the nonlocal Cauchy problem for semilinear differential inclusion, abstract integral inclusion of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusion.Existence and relaxation for subdifferential inclusions with unbounded perturbationhttps://www.zbmath.org/1483.340352022-05-16T20:40:13.078697Z"Timoshin, Sergey A."https://www.zbmath.org/authors/?q=ai:timoshin.sergey-aSummary: We consider a differential inclusion of subdifferential type with a nonconvex and unbounded valued perturbation. Existence and relaxation results are obtained for this inclusion. By relaxation we mean approximation of a solution of the differential inclusion with convexified perturbation by solutions of the given inclusion. The traditional condition of Lipschitz continuity for such kind of problems is weakened and a somehow more appropriate in the context of unbounded valued multifunctions ``truncated'' version of it is considered instead.Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theoryhttps://www.zbmath.org/1483.340372022-05-16T20:40:13.078697Z"Ao, Ji-jun"https://www.zbmath.org/authors/?q=ai:ao.jijun"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiongThe authors study the differential equation \[ (p u'')'' + q u = \lambda w u \] on \(J = (a, c) \cup (c, b)\) for finite \(a < c < b\), together with boundary conditions \[ A U (a) + B U (b) = 0, \quad U = \begin{pmatrix} u \\
u' \\
p u'' \\
(p u'')' \end{pmatrix}, \quad A, B \in M_4 (\mathbb{R}), \] and transmission conditions \[ C U (c-) + D U (c+) = 0. \] Here \(C, D\) are real-valued \(4 \times 4\)-matrices with positive determinants and the coefficient functions \(1/p, q, w\) are integrable on \(J\). Moreover, conditions on the matrices \(A, B\) are imposed which make the problem self-adjoint. As the main result, sufficient conditions are provided under which this eigenvalue problem is equivalent to a matrix eigenvalue problem of the form \((\mathbb P + \mathbb Q) \mathbb Y = \lambda \mathbb W \mathbb Y\), where \(\mathbb {P, Q, W}\) are constructed explicitly.
Reviewer: Jonathan Rohleder (Stockholm)Bound sets for a class of \(\varphi \)-Laplacian operatorshttps://www.zbmath.org/1483.340412022-05-16T20:40:13.078697Z"Feltrin, Guglielmo"https://www.zbmath.org/authors/?q=ai:feltrin.guglielmo"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioThe authors provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of \(\phi\)-Laplacian differential operators. Their main tool is a variant of the Manásevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. They also extend a classical theorem by Reissig for scalar periodically perturbed Liénard equations.
Reviewer: Alessandro Fonda (Trieste)Positive solutions to classes of infinite semipositone \((p,q)\)-Laplace problems with nonlinear boundary conditionshttps://www.zbmath.org/1483.340432022-05-16T20:40:13.078697Z"Sim, Inbo"https://www.zbmath.org/authors/?q=ai:sim.inbo"Son, Byungjae"https://www.zbmath.org/authors/?q=ai:son.byungjaeIn this interesting paper the authors study the existence, multiplicity and nonexistence of positive solutions for the one-dimensional \((p,q)\)-Laplacian problems:
\begin{gather*}
-(\varphi(u'))'=\lambda h(t)f(u),\qquad t\in (0,1),\\
u(0)=0=au'(1)+g(\lambda,u(1))u(1),
\end{gather*}
where \(\lambda>0\), \(a\geq 0\), \(\varphi(s)=|s|^{p-2}s+|s|^{q-2}s\) with \(1<p<q<+\infty\), \(h\in C((0,1),(0,+\infty))\), and \(f\in C((0,+\infty),\mathbb{R})\) may have a singularity at 0 of repulsive type. The proofs are based on a classical Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.
Reviewer: Manuel Zamora (Concepción)New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite intervalhttps://www.zbmath.org/1483.340442022-05-16T20:40:13.078697Z"Zhang, Wei"https://www.zbmath.org/authors/?q=ai:zhang.wei.10|zhang.wei.19|zhang.wei.6|zhang.wei.5|zhang.wei.3|zhang.wei.17|zhang.wei.15|zhang.wei.16|zhang.wei.9|zhang.wei.7|zhang.wei.1|zhang.wei.12|zhang.wei.4|zhang.wei.13|zhang.wei.2|zhang.wei.18"Ni, Jinbo"https://www.zbmath.org/authors/?q=ai:ni.jinboIn this paper, the authors consider nonlinear Hadamard-type fractional differential equations with nonlocal boundary conditions on an infinite interval. The existence of multiple positive solutions of the addressed problem is obtained by applying the generalized Avery-Henderson fixed point theorem. Finally, an example was given to show the effectiveness of the main result. This paper provides a new fixed point theorem to study multiple solutions.
Reviewer: Wengui Yang (Sanmenxia)On a series representation for integral kernels of transmutation operators for perturbed Bessel equationshttps://www.zbmath.org/1483.340462022-05-16T20:40:13.078697Z"Kravchenko, V. V."https://www.zbmath.org/authors/?q=ai:kravchenko.vladislav-v"Shishkina, E. L."https://www.zbmath.org/authors/?q=ai:shishkina.elina-leonidovna"Torba, S. M."https://www.zbmath.org/authors/?q=ai:torba.sergii-mSummary: A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.Periodic solutions for a class of \(n\)-dimensional prescribed mean curvature equationshttps://www.zbmath.org/1483.340572022-05-16T20:40:13.078697Z"Liang, Zai-tao"https://www.zbmath.org/authors/?q=ai:liang.zaitao"Lu, Shi-ping"https://www.zbmath.org/authors/?q=ai:lu.shipingIn this paper, there is investigated the periodic problem
\begin{align*}
\frac{d}{dt}\phi(x^{\prime})+\nabla W(x)&= p(t),\\ x(0)=x(T),\quad x^{\prime}(0)&= x^{\prime}(T),
\end{align*}
where \(x^\top=(x_1, x_2, \dots, x_n)\), \(W\in \mathcal{C}^1(\mathbb{R}^n)\), \(p\in \mathcal{C}(\mathbb{R})\), \(p(t+T)=p(t)\), \(\phi(x)= \frac{x}{\sqrt{1+|x|}}\). The authors provide conditions for the functions \(p\) and \(W\) so that the considered periodic problem has at least one periodic solution. The proof of the main result is based upon an extension of the Mawhin continuation theorem. The authors provide the main result with a suitable example
Reviewer: Svetlin Georgiev (Sofia)Global stability in a three-species Lotka-Volterra cooperation model with seasonal successionhttps://www.zbmath.org/1483.340702022-05-16T20:40:13.078697Z"Xie, Xizhuang"https://www.zbmath.org/authors/?q=ai:xie.xizhuang"Niu, Lin"https://www.zbmath.org/authors/?q=ai:niu.linSummary: In this paper, we focus on a three-species Lotka-Volterra cooperation model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. By Brouwer fixed point theorem and the connecting orbits theorem, it is proved that there admits a unique positive periodic solution under appropriate conditions. Furthermore, sharp global asymptotical stability criteria for extinction and coexistence are established. Compared to the classical three-species Lotka-Volterra cooperation model, the introduction of seasonal succession may lead to species' extinction. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.On the Lyapunov-Perron reducible Markovian master equationhttps://www.zbmath.org/1483.340802022-05-16T20:40:13.078697Z"Szczygielski, Krzysztof"https://www.zbmath.org/authors/?q=ai:szczygielski.krzysztofConvergence rates for boundedly regular systemshttps://www.zbmath.org/1483.340812022-05-16T20:40:13.078697Z"Csetnek, Ernö Robert"https://www.zbmath.org/authors/?q=ai:csetnek.erno-robert"Eberhard, Andrew"https://www.zbmath.org/authors/?q=ai:eberhard.andrew-c|eberhard.andrew"Tam, Matthew K."https://www.zbmath.org/authors/?q=ai:tam.matthew-kLet \(\mathcal{H}\) be a real Hilbert space; an operator \(T \colon \mathcal{H} \to \mathcal{H}\) is called Hölder regular on \(U \subseteq \mathcal{H}\) if there exist \(\kappa > 0\) and \(\gamma \in (0,1)\) such that
\[
d(y, \mathrm{Fix }\,T) \leq \kappa \parallel y - T(y) \parallel^\gamma \quad \forall y \in U.
\]
If \(T\) is Hölder regular on each bounded subset of \(\mathcal{H}\) then it is called boundedly Hölder regular. If the corresponding property is true for \(\gamma = 1,\) the operator \(T\) is called boundedly linearly regular.
The main result. Let \(T\) be nonexpansive with Fix\(\, T \neq \emptyset\) and \(\lambda \colon [0, +\infty) \to [0,1]\) be Lebesgue measurable function with \(\lambda^\star := \inf_{t\geq0} \lambda (t) >0.\) Let \(x\) be the unique strong global solution of the equation
\[
\dot{x}(t) = \lambda (t)(T(x(t)) - x(t)), \quad x(0) = x_0.
\]
If \(T\) is boundedly linearly regular, then there exists \(\overline{x} \in \mathrm{Fix }\,T\) such that for almost all \(t \in [0,+\infty)\) the following estimate holds:
\[
\parallel x(t) - \bar{x} \parallel \leq 2 \exp \Big(- \frac{\lambda^\star}{2\kappa^2}t\Big)\, d(x_0, \mathrm{Fix }\,T).
\]
An analogous result is also valid for the case of a boundedly Hölder regular operator \(T.\)
Reviewer: Valerii V. Obukhovskij (Voronezh)\((\omega,\mathbb{T})\)-periodic solutions of impulsive evolution equationshttps://www.zbmath.org/1483.340822022-05-16T20:40:13.078697Z"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"Liu, Kui"https://www.zbmath.org/authors/?q=ai:liu.kui"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrongSummary: In this paper, we study \((\omega,\mathbb{T})\)-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces \(X\), where \(\mathbb{T}: X\rightarrow X\) is a linear isomorphism. Existence and uniqueness of \((\omega,\mathbb{T})\)-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.Weak solvability for parabolic variational inclusions and application to quasi-variational problemshttps://www.zbmath.org/1483.340842022-05-16T20:40:13.078697Z"Kenmochi, Nobuyuki"https://www.zbmath.org/authors/?q=ai:kenmochi.nobuyuki"Niezgódka, Marek"https://www.zbmath.org/authors/?q=ai:niezgodka.marekThe authors investigate weak solvability of the class of parabolic variational evolution inclusions
\[
u^{\prime}(t)+\partial\varphi^{t}\left( p;u(t)\right) \ni f(t)\text{, }0<t<T
\]
\[
u(0)=u_{0}
\]
in a real Hilbert space, with nonlocal parameter \(p\), where \(\partial \varphi^{t}\left( p;\cdot\right) \) is the subdifferential of a time-dependent nonnegative convex function \(z\mapsto\varphi^{t}\left( p;z\right) \) with nonlocal dependence on \(p\), and \(f\in L^{2}\left( (0,T);H\right) \). \ The parameter \(p\) comes from a set \(X_{0}\) which is a bounded and closed subset of \(C\left( \left[ 0,T\right] ;X\right) \) where \(X\) is a real Banach space. \ Existence and uniqueness of a weak solution, and also continuous dependence on \(p\), \(f\) and \(u_{0}\), are proven under several technical assumptions. Specific examples are given for \(\varphi^{t}\left( p;\cdot\right) \). \ Existence of weak solutions for quasi-variational problems of the form
\[
u^{\prime}(t)+\partial\varphi^{t}\left( p;u(t)\right) \ni f(t)\text{, }0<t<T
\]
\[
u(0)=u_{0}
\]
\[
p=\Lambda_{p_{0}}u
\]
are also studied using these results and Schauder's fixed point theorem, where \(\Lambda_{p_{0}}\) is a feedback system which is an operator from a subset of \(C\left( \left[ 0,T\right] ;H\right) \) into \(X_{0}\). \ Weak solvability for perturbations of these problems is also considered and the paper closes with an application.
Reviewer: Daniel C. Biles (Nashville)Periodic solutions for some differential nonlinear systems with several delayshttps://www.zbmath.org/1483.340952022-05-16T20:40:13.078697Z"Gabsi, Hocine"https://www.zbmath.org/authors/?q=ai:gabsi.hocine"Ardjouni, Abdelouaheb"https://www.zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://www.zbmath.org/authors/?q=ai:djoudi.ahceneSummary: By means of continuation theorem of coincidence degree theory and Krasnoselskii-Burton's fixed point theorem we study some differential nonlinear systems of several delays with a deviating argument having the form \[\begin{cases}\frac{dx(t)}{dt}=\beta|x(t-\tau(t))|^\alpha x(t)+f(t,u(t-\sigma(t)))+p(t),\\ \frac{du(t)}{dt}=a(t)g(u(t))+G(t,x(t-\tau)),u(t-\sigma(t))),\end{cases}\] where \(\alpha\) and \(\beta\) are two parameters with \(0<\alpha<1\). We give sufficient conditions on \(\beta,\alpha,f,g\) and \(G\) to offer, what we hope, an existence criteria of periodic solutions of above system. Some new results on the existence of periodic solutions are obtained. We end by giving an example to illustrate our claim.On using coupled fixed-point theorems for mild solutions to coupled system of multipoint boundary value problems of nonlinear fractional hybrid pantograph differential equationshttps://www.zbmath.org/1483.341032022-05-16T20:40:13.078697Z"Iqbal, Muhammad"https://www.zbmath.org/authors/?q=ai:iqbal.muhammad-azhar|iqbal.muhammad-waqas|iqbal.muhammad-sajid|iqbal.muhammad-faisal|iqbal.muhammad-sohail|iqbal.muhammad-naveed|iqbal.muhammad-asad|iqbal.muhammad-zafar|iqbal.muhammad-kashif|iqbal.muhammad-javed|iqbal.muhammad-mutahir"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Khan, Rahmat Ali"https://www.zbmath.org/authors/?q=ai:khan.rahmat-aliIn this paper, authors consider a coupled systems of multipoint boundary value problems of fractional order hybrid differential equations. The perturbation is taken as nonliner and of second type. The fractional derivative is taken of Caputo's type. The unique solution of the boundary value problem is established under certain hypothesis. The fixed point technique is used to establish the results. Mainly Burton and couple type fixed point theorems are used. The proportional type delay that represent Pantograph equations is considered. Authors give two examples for better illustrations.
Reviewer: Syed Abbas (Mandi)Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equationshttps://www.zbmath.org/1483.341052022-05-16T20:40:13.078697Z"Ramos, Priscila Santos"https://www.zbmath.org/authors/?q=ai:ramos.priscila-santos"Sousa, J. Vanterler da C."https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"de Oliveira, E. Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: We discuss the existence and uniqueness of mild solutions for a class of quasi-linear fractional integro-differential equations with impulsive conditions via Hausdorff measures of noncompactness and fixed point theory in Banach space. Mild solution controllability is discussed for two particular cases.Faedo-Galerkin approximation of mild solutions of fractional functional differential equationshttps://www.zbmath.org/1483.341072022-05-16T20:40:13.078697Z"Vanterler da Costa Sousa, José"https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"de Oliveira, Edmundo Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.Controllability of impulsive fractional integro-differential evolution equationshttps://www.zbmath.org/1483.341092022-05-16T20:40:13.078697Z"Gou, Haide"https://www.zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://www.zbmath.org/authors/?q=ai:li.yongxiangSummary: In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term \(u'(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u'(b)=u'_b\). The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and \((\alpha,\beta)\)-resolvent operator. Finally, an example is given to illustrate the feasibility of our results.Existence, uniqueness and stability of fractional impulsive functional differential inclusionshttps://www.zbmath.org/1483.341102022-05-16T20:40:13.078697Z"da C. Sousa, J. Vanterler"https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Kucche, Kishor D."https://www.zbmath.org/authors/?q=ai:kucche.kishor-dSummary: In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the \(\psi \)-Liouville-Caputo fractional derivative, through fixed point theorem, Arzela-Ascoli theorem and multivalued analysis theory.Ulam's type stabilities for conformable fractional differential equations with delayhttps://www.zbmath.org/1483.341112022-05-16T20:40:13.078697Z"Wang, Sen"https://www.zbmath.org/authors/?q=ai:wang.sen"Jiang, Wei"https://www.zbmath.org/authors/?q=ai:jiang.wei.1"Sheng, Jiale"https://www.zbmath.org/authors/?q=ai:sheng.jiale"Li, Rui"https://www.zbmath.org/authors/?q=ai:li.rui.2|li.rui|li.rui.3|li.rui.1|li.rui.4Summary: In this paper, we investigate the existence and uniqueness of solutions and Ulam's type stabilities including the well-known Ulam-Hyers stability and the newly extended Ulam-Hyers conformable exponential stability for two classes of fractional differential equations with the conformable fractional derivative and the time delay. The Banach contraction principle, the technique of Picard operator, the Gronwall integral inequalities, and generalized iterated integral inequality in the sense of conformable fractional integral are the main tools for deriving our main results. Finally, several illustrative examples will be presented to demonstrate our work.Periodic solutions for a nonautonomous mathematical model of hematopoietic stem cell dynamicshttps://www.zbmath.org/1483.341142022-05-16T20:40:13.078697Z"Adimy, Mostafa"https://www.zbmath.org/authors/?q=ai:adimy.mostafa"Amster, Pablo"https://www.zbmath.org/authors/?q=ai:amster.pablo"Epstein, Julián"https://www.zbmath.org/authors/?q=ai:epstein.julianAuthors' abstract: The main purpose of this paper is to study the existence of periodic solutions for a nonautonomous differential-difference system describing the dynamics of hematopoietic stem cell (HSC) population under some external periodic regulatory factors at the cellular cycle level. The starting model is a nonautonomous system of two age-structured partial differential equations describing the HSC population in quiescent \((G_0)\) and proliferating (\(G_1\), \(S\), \(G_2\) and \(M\)) phase. We are interested in the effects of periodically time varying coefficients due for example to circadian rhythms or to the periodic use of certain drugs, on the dynamics of HSC population. The method of characteristics reduces the age-structured model to a nonautonomous differential-difference system. We prove under appropriate conditions on the parameters of the system, using topological degree techniques and fixed point methods, the existence of periodic solutions of our model.
Reviewer: Jiří Šremr (Brno)On the monotonicity of the best constant of Morrey's inequality in convex domainshttps://www.zbmath.org/1483.350072022-05-16T20:40:13.078697Z"Fărcăşeanu, Maria"https://www.zbmath.org/authors/?q=ai:farcaseanu.maria"Mihăilescu, Mihai"https://www.zbmath.org/authors/?q=ai:mihailescu.mihaiIn this interesting paper, the authors obtain some monotonicity properties of the best constant from Morrey's inequality in convex and bounded domains from the Euclidean space \(\mathbb{R}^{D}\) (\(D\ge1\)). Using these monotonicity properties, they give a new variational characterization of the best constant from Morrey's inequality on convex and bounded domains for which the maximum of the distance function to the boundary is small. The authors also show that this variational characterization does not hold true on convex and bounded domains for which the maximum of the distance function to the boundary is larger than one.
Reviewer: Meng Qu (Wuhu)Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operatorshttps://www.zbmath.org/1483.350122022-05-16T20:40:13.078697Z"Anh, Nguyen Thi Van"https://www.zbmath.org/authors/?q=ai:anh.nguyen-thi-van"Ke, Tran Dinh"https://www.zbmath.org/authors/?q=ai:ke.tran-dinh"Lan, Do"https://www.zbmath.org/authors/?q=ai:lan.doSummary: In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.Fractional oscillon equations; solvability and connection with classical oscillon equationshttps://www.zbmath.org/1483.350242022-05-16T20:40:13.078697Z"Bezerra, Flank D. M."https://www.zbmath.org/authors/?q=ai:bezerra.flank-david-morais"Figueroa-López, Rodiak N."https://www.zbmath.org/authors/?q=ai:figueroa-lopez.rodiak-n"Nascimento, Marcelo J. D."https://www.zbmath.org/authors/?q=ai:nascimento.marcelo-jose-diasSummary: In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation
\[
u_{tt} - \mu (t) \Delta u+ \omega (t)u_t = f(u),\, x \in \Omega,\, t \in \mathbb{R},
\]
subject to Dirichlet boundary condition on \(\partial \Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N \geq 3\), the function \(\omega\) is a time-dependent damping, \(\mu\) is a time-dependent squared speed of propagation, and \(f\) is a nonlinear functional. Under structural assumptions on \(\omega\) and \(\mu\) we establish the existence of time-dependent attractor for the fractional models in the sense of \textit{A. N. Carvalho} et al. [Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)], and \textit{F. Di Plinio} et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141--167 (2011; Zbl 1223.37100)].Decay rates for Kelvin-Voigt damped wave equations. II: The geometric control conditionhttps://www.zbmath.org/1483.350262022-05-16T20:40:13.078697Z"Burq, Nicolas"https://www.zbmath.org/authors/?q=ai:burq.nicolas"Sun, Chenmin"https://www.zbmath.org/authors/?q=ai:sun.chenminAuthors' abstract: ``We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coefficient is sufficiently smooth (C1 vanishing nicely, see the following equation: \(|\nabla a| \leq C a^{1/2}\)) we show that exponential decay follows from geometric control conditions (see [\textit{N. Burq} and \textit{H. Christianson}, Commun. Math. Phys. 336, No. 1, 101--130 (2015; Zbl 1320.35062); \textit{L. Tebou}, C. R., Math., Acad. Sci. Paris 350, No. 11--12, 603--608 (2012; Zbl 1255.35039)] for similar results under stronger assumptions on the damping function).''
For Part I, see [\textit{N. Burq}, SIAM J. Control Optim. 58, No. 4, 1893--1905 (2020; Zbl 1452.35030)].
Reviewer: Kaïs Ammari (Monastir)Generalized nonlocal Robin Laplacian on arbitrary domainshttps://www.zbmath.org/1483.350792022-05-16T20:40:13.078697Z"Oussaid, Nouhayla Ait"https://www.zbmath.org/authors/?q=ai:oussaid.nouhayla-ait"Akhlil, Khalid"https://www.zbmath.org/authors/?q=ai:akhlil.khalid"Aadi, Sultana Ben"https://www.zbmath.org/authors/?q=ai:ben-aadi.sultana"Ouali, Mourad El"https://www.zbmath.org/authors/?q=ai:el-ouali.mouradSummary: In this paper, we prove that it is always possible to define a realization of the Laplacian \(\Delta_{\kappa ,\theta }\) on \(L^2(\Omega )\) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of \({\mathbb{R}}^N\). This is made possible by using a capacity approach to define an admissible pair of measures \((\kappa ,\theta )\) that allows the associated form \({\mathcal{E}}_{\kappa ,\theta }\) to be closable. The nonlocal Robin Laplacian \(\Delta_{\kappa ,\theta }\) generates a sub-Markovian \(C_0\)-semigroup on \(L^2(\Omega )\) which is not dominated by the Neumann Laplacian semigroup unless the jump measure \(\theta\) vanishes.Magnetic Schrödinger operators with delta-type potentialshttps://www.zbmath.org/1483.350842022-05-16T20:40:13.078697Z"Rabinovich, Vladimir"https://www.zbmath.org/authors/?q=ai:rabinovich.vladimir-l|rabinovich.vladimir-sThe author considers a Schrödinger operator of the form
\[
H_{\rho,a,W}u(x)=\left[(-i\nabla+a(x))\rho(x)(-i\nabla+a(x)) +W(x)\right]u(x) \qquad\forall x\in {\mathbb{R}}^n\,,
\]
where \(\rho(x)\) is a symmetric \(n\times n\) matrix with real entries that is positive definite uniformly in \(x\in {\mathbb{R}}^n\) and bounded with its first order derivatives, \(a\) is a vector valued (magnetic) potential that is bounded with its first order derivatives, \(W\) is an essentially bounded complex-valued (electric) potential. Then the author introduces a hypersurface \(\Sigma\) of class \(C^2\) in \({\mathbb{R}}^n\) such that \({\mathbb{R}}^n\setminus \Sigma\) is the union of two domains \(\Omega_+\) and \(\Omega_-\) with the common boundary \(\Sigma\) and associates an unbounded operator \({\mathcal{H}}_{\rho,a,W,{\mathcal{A}}}\) in \(L^2({\mathbb{R}}^n)\) to a perturbation \(H_{\rho,a,W}+W_s\) of \(H_{\rho,a,W}\), where \(W_s\) is a singular potential with support in \(\Sigma\). Then the author proves that \({\mathcal{H}}_{\rho,a,W,{\mathcal{A}}}\) is self-adjoint and proves Fredholm properties of an operator associated to \(H_{\rho,a,W}+W_s\). Then the author exploits layer potentials associated to \(H_{\rho,a,W}\) to reduce the spectral problem
\[
(H_{\rho,a,W}+\lambda W_s)u =0\quad {\mathrm{for}}\quad u\in H^2(\Omega_+)\oplus H^2(\Omega_-)
\]
to the analysis of a corresponding problem for an operator \(A_\Sigma(\lambda)\) acting in \(H^s(\Sigma)\) for \(s\geq 0\) and proves that for certain subsets \(F\) of the complex plane, \(A_\Sigma(\lambda)\) is not invertible only for a discrete subset of \(\lambda\)'s in \(F\).
Reviewer: Massimo Lanza de Cristoforis (Padova)Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblemshttps://www.zbmath.org/1483.351432022-05-16T20:40:13.078697Z"Zhang, Ziyun"https://www.zbmath.org/authors/?q=ai:zhang.ziyunSummary: We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization using several examples, in particular a nonlinear Schrödinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.Strichartz estimates for Schrödinger equation with singular and time dependent potentials and application to NLS equationshttps://www.zbmath.org/1483.351702022-05-16T20:40:13.078697Z"Haque, Saikatul"https://www.zbmath.org/authors/?q=ai:haque.saikatulThe author establishes inhomogeneous Strichartz estimates for the Schrödinger equation
\[
i\partial_t u +\Delta u +Vu = F
\]
where \(V\) is a potential which is assumed to be in \(L^{d/2,\infty}(\mathbb{R}^d)\) and \(L^{\infty}\left(\mathbb{R}, L^{d/2,\infty}(\mathbb{R}^d)\right)\). In particular, the author shows the inhomogeneous Strichartz estimate
\[
||u||_{L^q L^r} \lesssim ||F||_{L^{\bar{q}'} L^{\bar{r}'}}
\]
holds for some non-admissible pairs of exponents. As an application, the author establishes existence of the scattering solution for the energy-critical focusing NLS with inverse-square potential.
Reviewer: Eric Stachura (Marietta)The essential spectrum of periodically stationary solutions of the complex Ginzburg-Landau equationhttps://www.zbmath.org/1483.352342022-05-16T20:40:13.078697Z"Zweck, John"https://www.zbmath.org/authors/?q=ai:zweck.john-w"Latushkin, Yuri"https://www.zbmath.org/authors/?q=ai:latushkin.yuri"Marzuola, Jeremy L."https://www.zbmath.org/authors/?q=ai:marzuola.jeremy-l"Jones, Christopher K. R. T."https://www.zbmath.org/authors/?q=ai:jones.christopher-k-r-tSummary: We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.Minnaert resonances for bubbles in soft elastic materialshttps://www.zbmath.org/1483.352542022-05-16T20:40:13.078697Z"Li, Hongjie"https://www.zbmath.org/authors/?q=ai:li.hongjie"Liu, Hongyu"https://www.zbmath.org/authors/?q=ai:liu.hongyu"Zou, Jun"https://www.zbmath.org/authors/?q=ai:zou.junAuthors' abstract: Minnaert resonance is a widely known acoustic phenomenon, and it has many important applications, in particular in the effective realization of acoustic metamaterials using bubbly media in recent years. In this paper, motivated by the Minnaert resonance in acoustics, we consider the low-frequency resonance for acoustic bubbles embedded in soft elastic materials. This is a hybrid physical process that couples the acoustic and elastic wave propagations. By delicately and subtly balancing the acoustic and elastic parameters as well as the geometry of the bubble, we show that Minnaert resonance can occur (at least approximately) for rather general constructions. Our study highlights the great potential for the effective realization of negative elastic materials by using bubbly elastic media.
Reviewer: Kaïs Ammari (Monastir)Null controllability of a degenerate cascade model in population dynamicshttps://www.zbmath.org/1483.352842022-05-16T20:40:13.078697Z"Echarroudi, Younes"https://www.zbmath.org/authors/?q=ai:echarroudi.younes"Maniar, Lahcen"https://www.zbmath.org/authors/?q=ai:maniar.lahcenSummary: This item is concerned with a controllability result of a population dynamics cascade model with one control force and two different scattering coefficients. Such coefficients are allowed to be null on the left-hand side of the gene-type domain. To this end, we follow the classical track consisting to prove the observability inequality for the full associated adjoint system using the so-called Carleman estimates and the semigroups theory.
For the entire collection see [Zbl 1476.34004].Melas-type bounds for the Heisenberg Laplacian on bounded domainshttps://www.zbmath.org/1483.353112022-05-16T20:40:13.078697Z"Kovařík, Hynek"https://www.zbmath.org/authors/?q=ai:kovarik.hynek"Ruszkowski, Bartosch"https://www.zbmath.org/authors/?q=ai:ruszkowski.bartosch"Weidl, Timo"https://www.zbmath.org/authors/?q=ai:weidl.timoSummary: We study Riesz means of the eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on bounded domains in \(\mathbb{R}^3\). We obtain an inequality with a sharp leading term and an additional lower order term, improving the result of \textit{A. M. Hansson} and \textit{A. Laptev} [Lond. Math. Soc. Lect. Note Ser. 354, 100--115 (2008; Zbl 1157.58008)].Approximation of the fractional Schrödinger propagator on compact manifoldshttps://www.zbmath.org/1483.353172022-05-16T20:40:13.078697Z"Chen, Jie Cheng"https://www.zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Da Shan"https://www.zbmath.org/authors/?q=ai:fan.dashan"Zhao, Fa You"https://www.zbmath.org/authors/?q=ai:zhao.fayouSummary: Let \(\mathcal{L}\) be a second order positive, elliptic differential operator that is self-adjoint with respect to some \(C^\infty\) density \(dx\) on a compact connected manifold \(\mathbb{M}\). We proved that if \(0<\alpha<1\), \(\alpha/2<s<\alpha\) and \(f\in H^s(\mathbb{M})\) then the fractional Schrödinger propagator \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}\) on \(\mathbb{M}\) satisfies \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}f(x)-f(x)=o(t^{s/\alpha-\varepsilon})\) almost everywhere as \(t\rightarrow 0^+\), for any \(\varepsilon>0\).Non-autonomous evolution equations of parabolic type with non-instantaneous impulseshttps://www.zbmath.org/1483.353332022-05-16T20:40:13.078697Z"Chen, Pengyu"https://www.zbmath.org/authors/?q=ai:chen.pengyu"Zhang, Xuping"https://www.zbmath.org/authors/?q=ai:zhang.xuping"Li, Yongxiang"https://www.zbmath.org/authors/?q=ai:li.yongxiangSummary: In this paper, we study the Cauchy problem to a class of non-autonomous evolution equations of parabolic type with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time \(t\) and generate an evolution family. New existence result of piecewise continuous mild solutions is established under more weaker conditions. At last, as a sample of application, the abstract result is applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.Spectral and ergodic properties of completely positive maps and decoherencehttps://www.zbmath.org/1483.370102022-05-16T20:40:13.078697Z"Fidaleo, Francesco"https://www.zbmath.org/authors/?q=ai:fidaleo.francesco"Ottomano, Federico"https://www.zbmath.org/authors/?q=ai:ottomano.federico"Rossi, Stefano"https://www.zbmath.org/authors/?q=ai:rossi.stefanoSummary: In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital \(C^\ast\)-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital \(\ast\)-endomorphisms of, in general non-unital, \(C^\ast\)-algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it into a \(C^\ast\)-algebra, and the map under consideration restricts to a unital \(\ast\)-automorphism for this new \(C^\ast\)-structure, thus generating a conservative dynamics on that persistent part.Koopman operators and the \(3x+1\)-dynamical systemhttps://www.zbmath.org/1483.370112022-05-16T20:40:13.078697Z"Leventides, John"https://www.zbmath.org/authors/?q=ai:leventides.john"Poulios, Costas"https://www.zbmath.org/authors/?q=ai:poulios.costasReiterative distributional chaos in non-autonomous discrete systemshttps://www.zbmath.org/1483.370312022-05-16T20:40:13.078697Z"Yin, Zongbin"https://www.zbmath.org/authors/?q=ai:yin.zongbin"Xiang, Qiaomin"https://www.zbmath.org/authors/?q=ai:xiang.qiaomin"Wu, Xinxing"https://www.zbmath.org/authors/?q=ai:wu.xinxingThe authors study several types of distributional chaos (DC) and reiterative distributional chaos (RDC) for discrete dynamical systems. They prove that RDC of type \(2\frac{1}{2}\) and type 2 are equivalent for linear operators on Banach spaces. The authors present a basic relation between RDC1 operators and RDC\(2\frac{1}{2}\) operators. Let \(f : X\rightarrow X\) be a continuous linear operator on a Banach space \(X\). It is proved that \(f\) is RDC1 if and only if it is RDC\(2\frac{1}{2}\).
Then, the authors investigate the iterative invariance of various types of RDC for nonautonomous discrete systems. Let \(f_{1,\infty} = \{f_i \}_{i\geq 1}\) be a sequence of self-maps of a metric space \(X\). It is proved that an equicontinuous nonautonomous system \((X,f_{1,\infty})\) exhibits RDC of type \(i\) with \(i\in \{1, 1+ , 2, 2\frac{1}{2},2\frac{1}{2}-\}\), if and only if its \(k\)-th iteration \(f^{[k]}_{1,\infty}\) exhibits RDC of type \(i\) for any \(k\geq 2\)
Reviewer: Hasan Akin (Trieste)Fixed points of the Ruelle-Thurston operator and the Cauchy transformhttps://www.zbmath.org/1483.370352022-05-16T20:40:13.078697Z"Levin, Genadi"https://www.zbmath.org/authors/?q=ai:levin.genadi-mSummary: We give necessary and sufficient conditions for a function in a naturally appearing function space to be a fixed point of the Ruelle-Thurston operator associated to a rational function (see Lemma 2.1). The proof uses essentially a 2020 paper of the author et al. [Nonlinearity 33, No. 8, 3970--4012 (2020; Zbl 1453.37044)]. As an immediate consequence, in Theorem 1 and Lemma 2.2 we revisit Theorem 1 and Lemma 5.2 of the author [in: Frontiers in complex dynamics. In celebration of John Milnor's 80th birthday. Based on a conference, Banff, Canada, February 2011. Princeton, NJ: Princeton University Press. 163--196 (2014; Zbl 1348.37075)].Noncommutative cross-ratio and Schwarz derivativehttps://www.zbmath.org/1483.370782022-05-16T20:40:13.078697Z"Retakh, Vladimir"https://www.zbmath.org/authors/?q=ai:retakh.vladimir-s"Rubtsov, Vladimir"https://www.zbmath.org/authors/?q=ai:rubtsov.vladimir-n"Sharygin, Georgy"https://www.zbmath.org/authors/?q=ai:sharygin.georgy-iSummary: We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to ``noncommutative elementary geometry'' and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated ``pentagramma mirificum''.
For the entire collection see [Zbl 1456.14004].Trace ideal properties of a class of integral operatorshttps://www.zbmath.org/1483.370862022-05-16T20:40:13.078697Z"Gesztesy, Fritz"https://www.zbmath.org/authors/?q=ai:gesztesy.fritz"Nichols, Roger"https://www.zbmath.org/authors/?q=ai:nichols.rogerSummary: We consider a particular class of integral operators \(T_{\gamma,\delta}\) in \(L^2(\mathbb{R}^n),n\in\mathbb{N},n\geqslant 2\), with integral kernels \(T_{\gamma, \delta}(\cdot ,\cdot)\) bounded (Lebesgue) a.e. by
\[
|T_{\gamma,\delta}(x,y)|\leqslant C\langle x\rangle^{-\delta}|x-y|^{2\gamma -n}\langle y\rangle^{-\delta}, \quad x,y\in\mathbb{R}^n, x\neq y,
\]
for fixed \(C\in(0,\infty),0 < 2\gamma < n,\delta > \gamma\), and prove that
\[
T_{\gamma,\delta}\in\mathcal{B}_p\big(L^2(\mathbb{R}^n)\big)\text{ for } p > n/(2\gamma), p\geqslant 2.
\]
(Here \(\langle x\rangle:=(1+|x|^2)^{1/2},x\in\mathbb{R}^n\), and \(\mathcal{B}_p\) abbreviates the \(\ell^p\)-based trace ideal.) These integral operators (and their matrix-valued analogs) naturally arise in the study of multi-dimensional Schrödinger and Dirac-type operators and we describe an application to the case of massless Dirac-type operators.
For the entire collection see [Zbl 1456.14003].Bi-additive s-functional inequalities and biderivation in modular spaceshttps://www.zbmath.org/1483.390122022-05-16T20:40:13.078697Z"Shateri, T. L."https://www.zbmath.org/authors/?q=ai:shateri.tayebe-laal|shateri.tayebeh-lalFor an algebra \(\mathcal{A}\) and a \(\rho\)-complete modular space \(\mathcal{X}_{\rho}\) (with a convex modular \(\rho\) satisfying some additional properties), the author considers a mapping
\[
d: \mathcal{A}\times \mathcal{A}\to\mathcal{X}_{\rho}
\]
which satisfies some conditions and inequalities involving \(d\), \(\rho\) and a control mapping \(\psi\). It is shown that under these conditions the mapping \(d\) can be approximated by a bi-additive mapping \(D\). If in addition some other conditions are satisfied, in particular if
\[
\rho(d(xy,z)-d(x,z)y-zd(y,z))\leq\psi(x,y)\psi(z,z),\quad x,y,z\in \mathcal{A},
\]
then \(d\) is a bi-derivation.
In the proofs, a fixed point method is used. Some particular forms of the control mapping \(\psi\) are considered as well.
Reviewer: Jacek Chmieliński (Kraków)Interactions between Hlawka type-1 and type-2 quantitieshttps://www.zbmath.org/1483.390132022-05-16T20:40:13.078697Z"Luo, Xin"https://www.zbmath.org/authors/?q=ai:luo.xinSummary: The classical Hlawka inequality possesses deep connections with zonotopes and zonoids in convex geometry, and has been related to Minkowski space. We introduce Hlawka Type-1 and Type-2 quantities, and establish a Hlawka-type relation between them, which connects a vast number of strikingly different variants of the Hlawka inequalities, such as Serre's reverse Hlawka inequality in the future cone of the Minkowski space, the Hlawka inequality for subadditive functions on abelian groups by \textit{P. Ressel} [J. Math. Inequal. 9, No. 3, 883--888 (2015; Zbl 1333.26009)], and the integral analogs by \textit{S.-E. Takahasi} et al. [Math. Inequal. Appl. 3, No. 1, 63--67 (2000; Zbl 0949.26012); Math. Inequal. Appl. 12, No. 1, 1--10 (2009; Zbl 1177.26050)]. Besides, we announce several enhanced results, such as the Hlawka inequality for the power of measure functions. Particularly, we give a complete study of the Hlawka inequality for quadratic forms which relates to a work of \textit{D. Serre} [C. R., Math., Acad. Sci. Paris 353, No. 7, 629--633 (2015; Zbl 1322.51002)].Stability property of functional equations in modular spaces: a fixed-point approachhttps://www.zbmath.org/1483.390142022-05-16T20:40:13.078697Z"Saha, P."https://www.zbmath.org/authors/?q=ai:saha.parbati"Mondal, Pratap"https://www.zbmath.org/authors/?q=ai:mondal.pratap"Choudhury, B. S."https://www.zbmath.org/authors/?q=ai:choudhury.binayak-samadderThis paper is devoted to the problem of stability of quadratic functional equations in modular metric spaces in the Hyers-Ulam-Rassias sense. More precisely, the functional equation here considered is the Pappus functional equation, which arises from some geometrical considerations. The obtained results generalize previously known findings about the considered equation.
Reviewer: Bilal Bilalov (Baku)On absolute Euler spaces and related matrix operatorshttps://www.zbmath.org/1483.400042022-05-16T20:40:13.078697Z"Gökçe, Fadime"https://www.zbmath.org/authors/?q=ai:gokce.fadime"Sarıgöl, Mehmet Ali"https://www.zbmath.org/authors/?q=ai:sarigol.mehmet-aliSummary: In the present paper, we extend Euler sequence spaces \(e_p^r\) and \(e_{\infty}^r\) by using the absolute Euler method in place of \(p\)-summable, which include the spaces \(l_p\), \(l_{\infty}\), \(e_p^r\) and \(e_{\infty}^r\), investigate some topological structures, and determine \(\alpha\)-, \(\beta\)-, \(\gamma\)-duals and base. Further, we characterize certain matrix and compact operators on those spaces, and also obtain their norms and Hausdorff measures of noncompactness.The subspace of almost convergent sequenceshttps://www.zbmath.org/1483.400052022-05-16T20:40:13.078697Z"Zvolinskii, R. E."https://www.zbmath.org/authors/?q=ai:zvolinskii.r-e"Semenov, E. M."https://www.zbmath.org/authors/?q=ai:semenov.evgueni-m|semenov.evgenii-mThe paper deals with a well-known summability method, called almost convergence method. The authors investigate the subspace of almost convergence sequences. They obtain some related theorems and convergence results using sine functions.
Reviewer: İsmail Aslan (Ankara)Reconstruction of two approximation processes in order to reproduce \(e^{ax}\) and \(e^{2ax}\), \(a>0\)https://www.zbmath.org/1483.410092022-05-16T20:40:13.078697Z"Yılmaz, Başar"https://www.zbmath.org/authors/?q=ai:yilmaz.basar"Uysal, Gümrah"https://www.zbmath.org/authors/?q=ai:uysal.gumrah"Aral, Ali"https://www.zbmath.org/authors/?q=ai:aral.aliThe authors proposed two modifications for Gauss-Weierstrass operators and moment-type operators which fix \(e^{ax}\) and \(e^{2ax}\), \(a>0\). They studied weighted approximation and proved Voronovskaya-type theorems in exponentially weighted spaces. Using the modulus of continuity in exponentially weighted spaces, they obtained some global smoothness preservation properties, and they gave a comparison result for Gauss-Weierstrass operators. Finally, they provided some graphical representations.
Reviewer: Naokant Deo (Delhi)Signal analysis using Born-Jordan-type distributionshttps://www.zbmath.org/1483.420042022-05-16T20:40:13.078697Z"Cordero, Elena"https://www.zbmath.org/authors/?q=ai:cordero.elena"de Gosson, Maurice"https://www.zbmath.org/authors/?q=ai:de-gosson.maurice-a"Dörfler, Monika"https://www.zbmath.org/authors/?q=ai:dorfler.monika"Nicola, Fabio"https://www.zbmath.org/authors/?q=ai:nicola.fabioThe Chapter contains results concerning recent advances in signal theory using time-frequency distributions. The authors demonstrate that some new members of the Cohen class that generalize the Wigner distribution are useful in damping artefacts in certain signals. Main properties as well as drawbacks are presented. Last but not least, several open problems are also discussed.
For the entire collection see [Zbl 1470.42002].
Reviewer: Liviu Goraş (Iaşi)Compactness of Riesz transform commutator associated with Bessel operatorshttps://www.zbmath.org/1483.420052022-05-16T20:40:13.078697Z"Duong, Xuan Thinh"https://www.zbmath.org/authors/?q=ai:duong.xuan-thinh"Li, Ji"https://www.zbmath.org/authors/?q=ai:li.ji.1"Mao, Suzhen"https://www.zbmath.org/authors/?q=ai:mao.suzhen"Wu, Huoxiong"https://www.zbmath.org/authors/?q=ai:wu.huoxiong"Yang, Dongyong"https://www.zbmath.org/authors/?q=ai:yang.dongyongSummary: Let \(\lambda >0\) and
\[
{\Delta_\lambda}: = - \frac{d^2}{dx^2} - \frac{2\lambda}{x}\frac{d}{dx}
\]
be the Bessel operator on \(\mathbb R_+:= (0,\infty)\). We first introduce and obtain an equivalent characterization of \(\mathrm{CMO}(\mathbb R_+, x^{2\lambda}dx)\). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function \(b\in\mathrm{BMO}(\mathbb R_+, x^{2\lambda}dx)\) is in \(\mathrm{CMO}(\mathbb R_+, x^{2\lambda}dx)\) if and only if the Riesz transform commutator \([b, R_{\triangle_\lambda}]\) is compact on \(L^p(\mathbb R_+, x^{2\lambda}dx)\) for all \(p\in (1,\infty)\).On the Gevrey wave front sethttps://www.zbmath.org/1483.420062022-05-16T20:40:13.078697Z"Kessab, Amor"https://www.zbmath.org/authors/?q=ai:kessab.amorSummary: We recall in this work some definitions of Gevrey's wave front set and we give a new definition using the Fourier-Bros Iagolnitzer (F.B.I) transform. We also show the invariance of this notion under the action of a Fourier integral operator.Boundedness of commutators of \(\theta\)-type Calderón-Zygmund operators on generalized weighted Morrey spaces over RD-spaceshttps://www.zbmath.org/1483.420092022-05-16T20:40:13.078697Z"Li, Qiumeng"https://www.zbmath.org/authors/?q=ai:li.qiumeng"Lin, Haibo"https://www.zbmath.org/authors/?q=ai:lin.haibo.1|lin.haibo"Wang, Xinyu"https://www.zbmath.org/authors/?q=ai:wang.xinyuIn this article the authors studied the boundedness of the commutators generated by the \(\theta\)-type Calderón-Zygmund operators and \(BMO\) functions on generalized weighted Morrey spaces over \(RD\)-spaces. By assuming slightly weaker conditions, the authors obtained the bounds for the above operators on the generalized weighted Morrey spaces \(\widetilde{\mathcal{M}}^{p,\psi}(w)\) and the generalized weighted Morrey spaces of \(L\ln L\) type \(\widetilde{\mathcal{M}}_{L\ln L}^{1,\psi}(w)\) over the \(RD\)-spaces.
Personally speaking, this paper is very interesting and very well written. This paper involves a large amount of definitions, notation and references, which increases its richness. Overall, this article is a nice piece of work.
Reviewer: Feng Liu (Qingdao)On families between the Hardy-Littlewood and spherical maximal functionshttps://www.zbmath.org/1483.420112022-05-16T20:40:13.078697Z"Dosidis, Georgios"https://www.zbmath.org/authors/?q=ai:dosidis.georgios"Grafakos, Loukas"https://www.zbmath.org/authors/?q=ai:grafakos.loukas\textit{E. M. Stein} [Proc. Natl. Acad. Sci. USA 73, 2174--2175 (1976; Zbl 0332.42018)] showed that the spherical maximal function
\[
S(f)(x) = \sup_{t>0} \frac1{\omega_{n-1}} \int_{\mathbb S^{n-1}} |f(x-t\theta)|d\sigma_{n-1}(\theta)
\]
is bounded from \(L^p(\mathbb R^n)\) to \(L^p(\mathbb R^n)\) when \(p>\frac{n}{n-1}\) and \(n\ge 3\), and is unbounded when \(p\le \frac{n}{n-1}\) and \(n\ge 2\). When \(n=2\) the positive direction of this result was proved by \textit{J. Bourgain} [J. Anal. Math. 47, 69--85 (1986; Zbl 0626.42012)]. The boundedness of the operator \(S\) was obtained via the auxiliary family of operators \(\{S_\alpha\},\, 0\le \alpha <1\), defined by
\[
S_\alpha(f)(x) = \sup_{t>0} \frac2{\omega_{n-1}B(\frac{n}2, 1-\alpha)} \int_{\mathbb B^n} |f(x-ty)|(1-|y|^2)^{-\alpha} dy,
\]
where \(B(x,y)\) is the Beta function. Another classical operator is the Hardy-Littlewood maximal function given by
\[
M(f)(x) = \sup_{t>0} \frac1{v_n} \int_{\mathbb B^n} |f(x-ty)| dy,
\]
where \(v_n\) is the volume of \(\mathbb B^n\). In the paper the authors study multilinear versions of \(S, \, S_\alpha\) and of \(M\). Specifically,
\[
M^m(f_1, \dots, f_m)(x) = \sup_{t>0} \frac1{v_{mn}}\int_{\mathbb B^{mn}} \prod_{i=1}^m|f_i(x-ty_i)|dy_1 \dots dy_m,
\]
with analogous definitions for \(S^m_\alpha(f_1, \dots, f_m)(x)\) and \(S^m(f_1,\dots,f_m)(x)\). The main result of the paper is the following. Theorem 2. Let \(0<\alpha<1\). Given \(f_i\in L^1_{loc}(\mathbb R^n)\) and \(x\in\mathbb R^n\) we have \begin{eqnarray*} &M^m(f_1, \dots, f_m)(x) \le S_\alpha^m(f_1,\dots,f_m)(x) \le S^m(f_1,\dots,f_m)(x),\\
&\lim_{\alpha \to 1^-} S_\alpha^m(f_1,\dots, f_m)(x) = S^m(f_1,\dots,f_m)(x),\\
&\lim_{\alpha\to 1^+ }S_\alpha^m(f_1,\dots,f_m)(x) = M^m(f_1, \dots, f_m)(x). \end{eqnarray*} These statements are valid even when some of the preceding expressions equal \(\infty\). Boundedness of the operators \(S_\alpha^m\) is considered in Theorem 3 where the authors prove the following. Let \(n\ge 2,\, 0\le\alpha <1\), and \(1< p_i\le \infty\). Define \(p\) by \(\sum_{i=1}^m \frac1{p_i} = \frac1{p}\). Then there is a constant \(C=C(m,\alpha, p_1,\dots,p_m)\) such that
\[
\|S_\alpha^m(f_1,\dots,f_m)\|_{L^p(\mathbb R^n)} \le C\prod_{i=1}^m \|f_i\|_{L^{p_i}(\mathbb R^n)}
\]
for all \(f_i\in L^{p_i}(\mathbb R^n)\) if and only if \(\frac{n}{mn-\alpha}<p\le \infty\).
Reviewer: Manfred Stoll (Columbia)Recent progress in bilinear decompositionshttps://www.zbmath.org/1483.420132022-05-16T20:40:13.078697Z"Fu, Xing"https://www.zbmath.org/authors/?q=ai:fu.xing"Chang, Der-Chen"https://www.zbmath.org/authors/?q=ai:chang.der-chen-e"Yang, Dachun"https://www.zbmath.org/authors/?q=ai:yang.dachunSummary: The targets of this article are twofold. The first one is to give a survey on bilinear decompositions for products of functions in Hardy spaces and their dual spaces, as well as their variants associated with the Schrödinger operator on Euclidean spaces. The second one is to give a new proof of the bilinear decomposition for products of functions in the Hardy space \(H^1\) and BMO on metric measure spaces of homogeneous type. Some applications to div-curl lemmas and commutators are also presented.Algebraic frames and dualityhttps://www.zbmath.org/1483.420182022-05-16T20:40:13.078697Z"Azadi, Shahrzad"https://www.zbmath.org/authors/?q=ai:azadi.shahrzad"Radjabalipour, Mehdi"https://www.zbmath.org/authors/?q=ai:radjabalipour.mehdiIn the present paper, authors discussed algebraic frames and their dual frames. It is shown that every algebraic frame has a dual frame. Few results regarding dual frames of algebraic frames are obtained. The authors also discussed Riesz-type algebraic frames which have unique algebraic dual frames and some results are given in this direction.
Reviewer: Virender Dalal (Delhi)The convergence constants and non linear approximations of fusion frameshttps://www.zbmath.org/1483.420222022-05-16T20:40:13.078697Z"Xu, Yuxiang"https://www.zbmath.org/authors/?q=ai:xu.yuxiang"Leng, Jinsong"https://www.zbmath.org/authors/?q=ai:leng.jinsongAuthors' abstract: In this paper, we study the unconditional constant and nonlinear N-term approximation of fusion frames in Hilbert spaces. We show that the unconditional constant and greedy constant are bounded by a constant which is associated with the fusion frame bounds. We prove that the unconditional constant of cross fusion frame expansions satisfies the similar properties as long as the cross g-frame expansions stay uniformly bounded away from zero. Finally, we show that fusion frames satisfy the quasi greedy and almost greedy conditions. Moreover, we prove that fusion Riesz bases satisfy the greedy condition.
Reviewer: Salvatore Ivan Trapasso (Torino)On a nonlinear integral equationhttps://www.zbmath.org/1483.450052022-05-16T20:40:13.078697Z"Maruyama, Toru"https://www.zbmath.org/authors/?q=ai:maruyama.toruSummary: The existence of a solution for a nonlinear integral equation of the form \((P)\) is proved based upon the Mazur-Hukuhara fixed point theory in locally convex spaces.On the existence of solutions for fractional integral equations by measure of non-compactness in Banach spacehttps://www.zbmath.org/1483.450072022-05-16T20:40:13.078697Z"Kazemi, Manochehr"https://www.zbmath.org/authors/?q=ai:kazemi.manochehrSummary: In this paper, the existence of the solutions of a class of fractional integral equations in aBanach algebra, are investigated. The main tools here are the technique of the measure of noncompactness and the Petryshyn's fixed point theorem. Also, for the applicability of the obtained results, some examples are given.On the pillars of functional analysishttps://www.zbmath.org/1483.460032022-05-16T20:40:13.078697Z"Velasco, M. Victoria"https://www.zbmath.org/authors/?q=ai:velasco.maria-victoriaSummary: Many authors consider that the main pillars of Functional Analysis are the Hahn-Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn's Lemma, while the latter two usually are obtained from Baire's Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire's theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.Bishop-Phelps-Bollobás property for positive operators when the domain is \(L_{\infty}\)https://www.zbmath.org/1483.460052022-05-16T20:40:13.078697Z"Acosta, María D."https://www.zbmath.org/authors/?q=ai:acosta.maria-d"Soleimani-Mourchehkhorti, Maryam"https://www.zbmath.org/authors/?q=ai:soleimani-mourchehkhorti.maryamIt has now been 14 years since the Bishop-Phelps-Bollobás property (BPBp, for short) was introduced and studied for the first time. Nowadays it seems to be still a very attractive field for several researchers. In the present paper, the authors consider this property for positive operators between two (real) Banach lattices. It was known previously that the pairs \((c_0, L_1(\nu))\) and \((L_{\infty}(\mu), L_1(\nu))\) have the BPBp for positive operators for any positive measures \(\mu\) and \(\nu\). Now, the authors extend these results and prove that \((c_0, Y)\) has the BPBp for positive operators whenever \(Y\) is a uniformly monotone Banach lattice as well as \((L_{\infty}(\mu), Y)\) does for any positive measure \(\mu\) and \(Y\) being a uniformly monotone Banach lattice with a weak unit. It is worth remarking that there exists a Banach function space \(Y\) such that \((c_0, Y)\) fails the BPBp for positive operators and that it is not known whether the pair \((c_0, \ell_1)\) has the BPBp for operators in the real case.
Reviewer: Sheldon Dantas (Castelló)The factorisation property of \(\ell^\infty(X_k)\)https://www.zbmath.org/1483.460142022-05-16T20:40:13.078697Z"Lechner, Richard"https://www.zbmath.org/authors/?q=ai:lechner.richard"Motakis, Pavlos"https://www.zbmath.org/authors/?q=ai:motakis.pavlos"Müller, Paul F. X."https://www.zbmath.org/authors/?q=ai:muller.paul-f-x"Schlumprecht, Thomas"https://www.zbmath.org/authors/?q=ai:schlumprecht.thomasSummary: In this paper we consider the following problem: let \(X_k\) be a Banach space with a normalised basis \((e_{(k, j)})_j\), whose biorthogonals are denoted by \((e_{(k,j)}^*)_j\), for \(k\in\mathbb{N}\), let \(Z=\ell^\infty(X_k:k\in\mathbb{N})\) be their \(\ell^\infty\)-sum, and let \(T:Z\to Z\) be a bounded linear operator with a large diagonal, i.e.,
\[
\inf\limits_{k,j}\left|e^*_{(k,j)}(T(e_{(k,j)})\right|>0.
\]
Under which condition does the identity on \(Z\) factor through \(T\)? The purpose of this paper is to formulate general conditions for which the answer is positive.RKH spaces of Brownian type defined by Cesàro-Hardy operatorshttps://www.zbmath.org/1483.460232022-05-16T20:40:13.078697Z"Galé, José E."https://www.zbmath.org/authors/?q=ai:gale.jose-e"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-j"Sánchez-Lajusticia, Luis"https://www.zbmath.org/authors/?q=ai:sanchez-lajusticia.luisSummary: We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.Measure of noncompactness of Sobolev embeddings on strip-like domainshttps://www.zbmath.org/1483.460292022-05-16T20:40:13.078697Z"Edmunds, David E."https://www.zbmath.org/authors/?q=ai:edmunds.david-eric"Lang, Jan"https://www.zbmath.org/authors/?q=ai:lang.jan"Mihula, Zdeněk"https://www.zbmath.org/authors/?q=ai:mihula.zdenekThis paper is devoted to the study of the Sobolev embeddings \(W^{1,p}_0(D)\to L^p(D)\), where \(p\in(0,\infty)\) and \(D\) is a strip-like domain of the form \(\mathbb{R}^k\times \prod_{j=1}^{n-k}(r_j,q_j)\) with \(n\ge 2\), \(1\le k\le n-1\) and \(0<q_j<r_j<\infty\). Via computing the precise value of the measure of noncompactness \(\beta(I_p)\) of the natural embeddings \(I_p:\ W^{1,p}_0(D)\to L^p(D)\) as \[\beta(I_p)=\left(1+(p-1) \left(\frac{2\pi}{p\sin(\pi/p)}\right)^p\sum_{j=1}^{n-k}(r_j-q_j)^{-p}\right)^{-1/p}\] and proving the coincidence \(\beta(I_p)=\|I_p\|\), it is proved that the embedding \(I_p\) is maximally noncompact. Besides, the authors show that not only the measure of noncompactness but also all strict \(s\)-numbers of the embeddings in question coincide with their norms. Furthermore, the maximal noncompactness of Sobolev embeddings on strip-like domains is also proved to be valid even when Sobolev-type spaces are built upon general rearrangement-invariant spaces. As a by-product, the explicit form for the first eigenfunction of the pseudo-\(p\)-Laplacian on an \(n\)-dimensional rectangle is also obtained.
Reviewer: Wen Yuan (Beijing)Maximal non-compactness of Sobolev embeddingshttps://www.zbmath.org/1483.460322022-05-16T20:40:13.078697Z"Lang, Jan"https://www.zbmath.org/authors/?q=ai:lang.jan"Musil, Vít"https://www.zbmath.org/authors/?q=ai:musil.vit"Olšák, Miroslav"https://www.zbmath.org/authors/?q=ai:olsak.miroslav"Pick, Luboš"https://www.zbmath.org/authors/?q=ai:pick.lubosSummary: It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces \(L^{p,\infty}\) has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as \(p<\infty\). Finally, we show that if the target space is \(L^{\infty}\) (which formally is also a weak Lebesgue space with \(p=\infty)\), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.A framework of linear canonical Hankel transform pairs in distribution spaces and their applicationshttps://www.zbmath.org/1483.460402022-05-16T20:40:13.078697Z"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohan"Kumar, Manish"https://www.zbmath.org/authors/?q=ai:kumar.manish.1"Pradhan, Tusharakanta"https://www.zbmath.org/authors/?q=ai:pradhan.tusharakantaSummary: The motivation of this article stems from the fact that weak solutions of some partial differential equations exist in a distributional sense, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. For \(1\leqq p<\infty\) and \(s\in\mathbb{R}\), we have introduced a new definition for each of the following Sobolev-type spaces:
\[
\mathscr{W}^{s ,p, \mathscr{M}}_{1, \mu ,\nu ,\alpha, \beta}(I) \qquad \text{and} \qquad \mathscr{W}^{s, p, \mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I)
\]
as subspaces of
\[
H^{'\mathscr{M}}_{1, \mu, \nu, \alpha, \beta}(I) \qquad \text{and} \qquad H^{'\mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I),
\]
respectively, by using a linear canonical Hankel transform pair, where \(\mu\), \(\nu\), \(\alpha\) and \(\beta\) are real parameters and \(\mathscr{M}\) is a \(2\times 2\) real (or complex) matrix with determinant equal to 1. Any \(f\in H^{'\mathscr{M}}_{1, \mu, \nu, \alpha, \beta}(I)\) and \(g\in H^{'\mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I)\) with compact support are shown to be an element of the spaces:
\[
\mathscr{W}^{s ,p, \mathscr{M}}_{1, \mu, \nu, \alpha, \beta}(I) \qquad \text{and} \qquad\mathscr{W}^{s, p, \mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I),
\]
respectively, for the large negative value of \(s\). Examples in these spaces are constructed and the corresponding solutions are obtained. We have shown that these spaces turn out to be Hilbert spaces with respect to a certain norm with the dual spaces:
\[
\mathscr{W}^{-s, p, \mathscr{M}}_{1, \mu, \nu, \alpha, \beta}(I) \qquad \text{and} \qquad \mathscr{W}^{-s, p, \mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I),
\]
respectively. Further, if \(f\in\mathscr{W}^{s, p, \mathscr{M}}_{1, \mu, \nu, \alpha, \beta}(I)\), then \(x^{-\nu\mu + \alpha - 2\nu + 1}f(x)\) is shown to be bounded. Similarly, if \(g\in\mathscr{W}^{s, p, \mathscr{M}}_{2, \mu, \nu, \alpha, \beta}(I)\), then \(x^{-\nu\mu - \alpha}g(x)\) is also shown to be bounded. Furthermore, some applications of linear canonical Hankel transform pairs are provided in order to solve some generalized non-homogeneous partial differential equations. Finally, in the concluding section, some motivations and directions are indicated for further researches related to the areas which are considered and discussed in this article.Spectral homomorphisms on a locally convex algebra \(C_b(X)\)https://www.zbmath.org/1483.460442022-05-16T20:40:13.078697Z"Nowak, Marian"https://www.zbmath.org/authors/?q=ai:nowak.marianSummary: Let \(X\) be a completely regular Hausdorff space and \(\mathcal{B}_0\) be the \(\sigma\)-algebra of Borel sets in \(X\). Then the space \(C_b(X)\) (resp. \(B(\mathcal{B}_0)\)) of all bounded continuous (resp. bounded \(\mathcal{B}_0\)-measurable) complex functions on \(X\), equipped with the natural strict topology \(\beta\) is a locally convex algebra with the jointly continuous multiplication. It is shown that every \((\beta,\xi)\)-continuous homomorphism from \(C_b(X)\) to a complex sequentially complete locally convex algebra \((A,\xi)\) that maps \(1\!\!1_X\) to a unit 1 in \(\mathcal{A}\) is a spectral homomorphism for a unique spectral measure \(m:\mathcal{B}_0\rightarrow\mathcal{A}\). As an application, we study continuous algebra homomorphisms from \((C_b(X),\beta)\) to the algebra \(\mathcal{L}(F)\) of all bounded linear operators on a Banach space \(F\), equipped with the strong operator topology.Pietsch's variants of \(s\)-numbers for multilinear operatorshttps://www.zbmath.org/1483.460452022-05-16T20:40:13.078697Z"Fernandez, D. L."https://www.zbmath.org/authors/?q=ai:fernandez.dicesar-lass"Mastyło, M."https://www.zbmath.org/authors/?q=ai:mastylo.mieczyslaw"Silva, E. B."https://www.zbmath.org/authors/?q=ai:silva.eduardo-brandaniIn this paper the authors extend Pietsch's theory of $s$-numbers of operators to multilinear mappings. Given a natural number $k$ they say that an $s^{(k)}$-scale is a rule which assigns to each $k$-linear operator, $T$, from Banach spaces $X_1\times\cdots\times X_k$ to $F$ a sequence of non-negative numbers $(s_n^{(k)})_n$ such that
(S1) For each $k$-linear mapping $T$ $$ \|T\|=s_1^{(k)}(T)\ge s_2^{(k)}(T)\ge\cdots\ge 0. $$ (S2) For every pair of $k$-linear mappings $S,T\colon X_1\times\cdots\times X_k\to F$, $$ s_{m+n-1}^{(k)}(S+T)\le s_m^{(k)}(S)+s_n^{(k)}(T). $$ (S3) Given a $k$-linear mapping $T\colon X_1\times\cdots\times X_k\to F$ and linear mappings $S\colon Y\to Z$, $R_j\colon W_j\to X_j$, $1\le j\le k$, $$ s_n^{(k)}(S\circ T\circ(R_1,\ldots, R_k))\le \|S\|s_n^{(k)}(T)\|R_1\|\dots \|R_k\|. $$ (S4) If $\mathrm{rank}(T)$ (defined as the dimension of the span of $T(X_1,\ldots, X_k)$) is strictly less than $n$, then $S_n^{(k)}(T)=0$.
(S5) If $\bigotimes_kI_n$ is the $k$-linear mapping from $\underbrace{\ell_2^n \times\cdots\times \ell_2^n}_{k\text{-times}}$ to $\ell_2^{[n]^k}$ given by
$$
\bigotimes_k I_n( x_1,\ldots, x_k)(\mathbf{j})=x_1(j_1)\ldots x_k(j_k) \ \text{for} \ \mathbf{j}=(j_1,\ldots,j_k)\in [n]^k,
$$
then $s_n^{(k)}(\bigotimes_k I_n)=1$.
An $s^{(k)}$-scale is said to be injective if $s_n^{(k)}(T)=s_n^{(k)}(JT)$ for all $k$-linear mappings $T\colon X_1\times\ldots\times X_k\to Y$ and all metric injections $J\colon Y\to Z$, and surjective if $ s_n^{(k)}(T) =s_n^{(k)}(T(Q_1,\ldots,Q_k))$ for all $k$-linear mappings $T\colon X_1\times\ldots\times X_k\to Y$ and all metric surjections $Q_j\colon Y_j \to X_j$, $1\le j\le k$.
The authors define the $n$-th approximation number of a $k$-linear mapping $T\colon X_1\times\ldots\times X_k\to Y$, $a_n^{(k)}$, by $$ a_n^{(k)}(T)=\inf\{\|T-A\|: A\colon X_1\times\ldots\times X_k\to Y, \ k\text{-linear with rank}(A)<n\}, $$ the $n$-th Gelfand number, of a $k$-linear mapping $T\colon X_1\times \ldots\times X_k\to Y$, $c_n^{(k)}$, by $c_n^{(k)}=a_n^{(k)}(J_YT)$, where $J_Y\colon Y\to \ell_\infty(B_{Y^*})$ is the canonical injection given by $J_Y(y)=(y^*(y))_{y^*\in B_{ Y^*}}$, and the $n$-th Kolmogorov number of a $k$-linear mapping $T\colon X_1\times\ldots\times X_k\to Y$, $d_n^{(k)}$, by $d_n^{(k)}=a_n^{(k)}(T(Q_1, \ldots, Q_k))$, where $Q_j$ is the canonical metric surjection from $\ell_1 (B_{X_j})$ onto $X_j$. They show that $(a_n^{(k)})_n$ is the largest sequence of $s^{(k)}$-scales, $(c_n^{(k)})_n$ is the largest injective sequence of $s^{(k)}$-scales and $(d_n^{(k)})_n$ is the largest surjective sequence of $s^{(k)}$-scales. Moreover, for each decreasing sequence $(\lambda_n)_n$ of strictly positive real numbers there is a diagonal bilinear mapping $D_\lambda\colon\ell_p\times\ell_p\to \ell_p$ with $a_r^{(2)}(D_\lambda) =\lambda_r$ for each $r$ in $\mathbb{N}$. In an analogous way to scales of linear operators, the $s^{(k)}$-scales of Hilbert, Weyl and Chang numbers are defined using linear operators from and to $\ell_2$. The paper concludes with the introduction of the Bernstein numbers for bilinear operators and an inspection of which of the properties of an $s^{(2)}$-scale they possess.
Reviewer: Christopher Boyd (Dublin)On some applications of representable and continuous functionals of Banach quasi \(^\ast \)-algebrashttps://www.zbmath.org/1483.460522022-05-16T20:40:13.078697Z"Adamo, Maria Stella"https://www.zbmath.org/authors/?q=ai:adamo.maria-stellaSummary: This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi \(^\ast \)-algebras. In particular, we look at the link between the notions of \(^\ast \)-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying \(^\ast \)-derivations and representability properties for the tensor product of Hilbert quasi \(^\ast \)-algebras, a special class of Banach quasi \(^\ast \)-algebras.
For the entire collection see [Zbl 1471.47002].Topological algebras -- geometry -- physics; some interactionshttps://www.zbmath.org/1483.460532022-05-16T20:40:13.078697Z"Fragoulopoulou, Maria"https://www.zbmath.org/authors/?q=ai:fragoulopoulou.mariaSummary: This is a survey account on some interactions among topological algebras, geometry and physics.
For the entire collection see [Zbl 1466.46001].Unbounded local completely positive maps of local order zerohttps://www.zbmath.org/1483.460592022-05-16T20:40:13.078697Z"Joiţa, Maria"https://www.zbmath.org/authors/?q=ai:joita.mariaMotivated by \textit{A. Dosiev} [J. Funct. Anal. 255, No. 7, 1724--1760 (2008; Zbl 1198.46044)], the author introduces the notion of local orthogonality in a locally \(C^*\)-algebra. She also describes the structure of an unbounded operator valued local completely contractive and local completely positive map preserving the local orthogonality.
Reviewer: Maryam Amyari (Mashhad)Crossed-products extensions, of \(L_p\)-bounds for amenable actionshttps://www.zbmath.org/1483.460652022-05-16T20:40:13.078697Z"González-Pérez, Adrián M."https://www.zbmath.org/authors/?q=ai:gonzalez-perez.adrian-manuelSummary: We will extend earlier transference results due to \textit{S. Neuwirth} and \textit{É. Ricard} [Can. J. Math. 63, No. 5, 1161--1187 (2011; Zbl 1251.47036)] from the context of noncommutative \(L_p\)-spaces associated with amenable groups to that of noncommutative \(L_p\)-spaces associated with crossed-products of amenable actions. Namely, if \(m : G \rightarrow \mathbb{C}\) is a completely bounded Fourier multiplier on \(L_p\), then it extends to the crossed-product with similar bounds provided that the action \(\theta\) is amenable and trace-preserving. Furthermore, our construction also allows to extend \(G\)-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Følner sets of the action \(\theta\) satisfy certain accretivity property. As a corollary, we obtain stability results for maximal \(L_p\)-bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the \(L_p\)-spaces of group algebras.Energy-constrained diamond norms and quantum dynamical semigroupshttps://www.zbmath.org/1483.460722022-05-16T20:40:13.078697Z"Shirokov, M. E."https://www.zbmath.org/authors/?q=ai:shirokov.maksim-evgenevich"Holevo, A. S."https://www.zbmath.org/authors/?q=ai:holevo.alexander-sSummary: In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as an exponential series converging w.r.t. the energy-constrained diamond norm. Examples of semigroups having such a representation are presented.Quantum tomography and the quantum Radon transformhttps://www.zbmath.org/1483.460732022-05-16T20:40:13.078697Z"Ibort, Alberto"https://www.zbmath.org/authors/?q=ai:ibort.alberto"López-Yela, Alberto"https://www.zbmath.org/authors/?q=ai:lopez-yela.albertoSummary: A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of \(C^\ast\)-algebras is presented. Given a \(C^\ast\)-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on \(C^\ast\)-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.
The abstract theory is realized by using dynamical systems, that is, groups represented on \(C^\ast\)-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judicious use of the theory of frames. A few significant examples are discussed that illustrate the use and scope of the theory.Perturbation theory for linear operators. Denseness and bases with applicationshttps://www.zbmath.org/1483.470012022-05-16T20:40:13.078697Z"Jeribi, Aref"https://www.zbmath.org/authors/?q=ai:jeribi.arefThis book attends to some recent development of perturbation theory of non-self-adjoint linear operators, in particular, to the completeness and basis property of the generalised eigenvectors (or root vectors) of discrete operators in Hilbert spaces or Banach spaces. For some concrete models coming from physical problems, the asymptotic behaviour of eigenvalues is also discussed. By the perturbation method, the structural properties of these models are derived.
The book consists of fifteen chapters. The first six chapters cover basic material from functional analysis and operator theory that underlies most of the concepts used in this book. Especially, Chapter~6 collects some results on the completeness of generalised eigenvectors (or root vectors) of discrete operators and gives some criteria on the completeness of generalised eigenvectors of unbounded linear operators with compact resolvent in separable Hilbert spaces.
Chapter 7 concerns bases of separable Banach spaces, the Schauder basis for Banach spaces, and orthonormal bases for Hilbert spaces, in particular, equivalent bases on Banach spaces or Hilbert spaces. The equivalence of bases is an important concept and also is a principal tool of bases perturbation, with which some important bases such as Riesz basis and Riesz basis of Jordan chains in Hilbert spaces are introduced. As an extension of Riesz basis, the Riesz bases of subspaces, finitely spectral basis related to closed linear operators, Riesz basis with parentheses, and L-Riesz basis of exponentials for \(L_2[0,T]\) are recalled.
Chapter 8 is devoted to the study of the Riesz basis of finite-dimensional invariant subspaces for a class of unbounded perturbations of unbounded normal operators. The perturbed operator is of the form \(T=G+S\) where the major operator \(G\) is an unbounded normal operator with compact resolvent whose spectrum lies on finitely many rays from the origin, and the perturbation operator \(S\) is \(G\)-bounded with order \(p\in(0,1)\). The spectral property of \(T\) is discussed. Under the separable condition of eigenvalues (or spectral gap condition) of \(G\), the Riesz basis property of root subspaces of \(T\) is obtained, in which it is called Riesz basis of finite-dimensional invariant subspaces. The results are applied to two classes of block operator matrices. In particular, using the \(L\)-Riesz basis of the exponentials, the existence of a Riesz basis with parentheses of generalised eigenvectors for the generator of a \(C_0\)-semigroup in Hilbert space is obtained via the semigroup method, using the result and method from \textit{G. Q. Xu} and \textit{S. P. Yung}'s paper [J. Math. Anal. Appl. 328, No. 1, 245--256 (2007; Zbl 1110.47034)].
Chapter 9 covers some supplements of the perturbation theory of linear operators by considering an \(\ell^1\) non-analytic perturbation by what is termed analytic operators in Feki-Jeribi-Sfaxi's sense. More precisely, let \(T_0\) be a closed and densely defined linear operator in \(X\) with discrete eigenvalues and \(T_k\) be \(T_0\)-bounded linear operators with order \(p\in (0,1)\) and satisfying, for \(q>0\),
\[\|T_k\varphi \|\leq q^k(a\|T_0\varphi \|^p\|x\|^{1-p}+b\|\varphi \|),\ \varphi \in D(T_0) \text{ for all }k\ge 1.\]
Let \(\xi=\{\xi_k\}^\infty_{k=1}\in\ell^1\). The operator
\[T(\xi)\varphi=T_0\varphi+\sum^\infty_{k=1}\xi_kT_k \varphi \]
is said to be a non-analytic perturbation of \(T_0\). If \(\xi_k=\varepsilon^k\), it is said to be an analytic perturbation. Analogues to classic perturbation, the closedness of the operator \(T(\xi)\), the resolvent series expression, the relation of eigenvalues and associated eigenvectors between \(T_0\) and \(T(\xi)\), as well as series expansion are discussed. Here we mention that, in the proofs, the condition \(\|\varphi_n\|=\|\varphi^*_n\|\) and \(\varphi^*_n(\varphi_n)=1\) plays an essential role.
Chapter 10 is devoted to study the basis property of analytic operators under certain conditions and extends Keldysh's theorem for self-adjoint operators [M.V. Keldysh, ``On eigenvalues and eigenfunctions of some classes of non-self-adjoint equations'' (in Russian), Dokl. Akad. Nauk SSSR 77, 11--14 (1951; Zbl 0045.39402)] to the case of more general operators. More precisely, \(X\) is a Hilbert space, \(T_0\) is a unbounded operator with resolvent belonging to the Carleman-classs \(\ell^p\) and each eigenvalue is simple, and \(T_k\) are \(T_0\)-compact. The completeness of root vectors of \(T(\varepsilon)\) is proved. Furthermore, if the eigenvector of \(T_0\) forms a Riesz basis for \(X\), then there is a sequence of \(\varepsilon_n\) such that \(\{\varphi_n(\varepsilon)\}\) forms a Riesz basis for \(X\). In particular, if \(T_0\) is a normal operator whose spectrum lies on finitely many rays, then \(T(\varepsilon)\) has a subspace Riesz basis of eigenspaces. As a special case, for an operator \(T_0\) with the property that \(\{e^{\lambda_n t}\}\) forms a Riesz basis for some \(L_2(0,T)\), it is proved that \(\{e^{\lambda_n(\xi) t}\}\) is also a Riesz basis for \(L_2(0,T)\) provided that the non-analytic operator \(T(\xi)\) satisfies a certain smallness. Here, we mention that the uniform spectral gap condition is needed in the proofs.
Chapters 11 and 12 study the spectral property of an analytic operator \(T(\varepsilon)\), in which \(T_0\) is a self-adjoint and positive operator with compact resolvent satisfying the spectral ``not condense'' condition, and \(T_k\) are \(T_0\) bounded with order \(\beta\in (0,1)\). It is proved that, if \(\beta\in (0,\frac{1}{2})\), \(T(\varepsilon)-T_0\) is \(T_0\)-compact, and hence \(T(\varepsilon)\) has discrete spectrum for small \(\varepsilon\). Estimations of the location and density of the eigenvalues of \(T(\varepsilon)\) are given in terms of \(T_0\). In addition, based on the localisation of the eigenvalues of \(T(\varepsilon)\), the uniform boundedness of partial sums of the Riesz spectral project is proved, and hence the Riesz basis with parentheses of the root vector of the analytic operator is obtained. The ideas of the proofs are mainly from work of \textit{A. A. Shkalikov} [Proc. Steklov Inst. Math. 269, 284--298 (2010; Zbl 1200.47021); translation from Trudy Mat. Inst. Steklova 269, 290--303 (2010)].
The last three chapters concentrate on a selection of applications to a perturbation method for sound radiation by a vibrating plate in a light fluid, Gribov operator in Bargmann space, and also applications in mathematical physics and mechanics. Chapter~13 applies the perturbation method to analyse spectral properties of operators governing sound radiation by a vibrating plate in a light fluid and shows compactness of the resolvent operator, completeness of the generalised eigenvectors, as well as existence of Riesz basis or basis with parentheses. Chapter~14 applies the perturbation method to the Gribov operator in Bargmann space and confirms the existence of a Riesz basis. Chapter~15 contains some applications in mathematical physics and mechanics to investigate the expansion of solutions according to generalised eigenvectors for a rectilinear transport equation and the Lamé system, in which the generalised eigenvectors fail to have the basis property, while using the spectral information of operator, the expansion property of solution for some \(t\ge h\) is still true.
This book provides a very good collection of results for the study of the structural property of unbounded linear operators with compact resolvent, in particular, for the study of non-selfadjoint operators in Hilbert spaces. Due to many typos, readers and researchers should read it carefully.
Reviewer: Gen Qi Xu (Tianjin)Operators with compatible rangeshttps://www.zbmath.org/1483.470022022-05-16T20:40:13.078697Z"Djikić, Marko S."https://www.zbmath.org/authors/?q=ai:djikic.marko-sSummary: A bounded operator \(T\) on a finite or infinite-dimensional Hilbert space is called a disjoint range (DR) operator if \(\mathcal{R}(T) \cap \mathcal{R}(T^\ast)=\{0\}\), where \(T^\ast\) stands for the adjoint of \(T\), while \(\mathcal{R}(\cdot)\) denotes the range of an operator. Such operators (matrices) were introduced and systematically studied by \textit{O. M. Baksalary} and \textit{G. Trenkler} [Linear Algebra Appl. 435, No. 6, 1222--1240 (2011; Zbl 1221.15005)], and later by \textit{C.-Y. Deng} et al. [ibid. 437, No. 9, 2366--2385 (2012; Zbl 1276.47004)]. In this paper, we introduce a wider class of operators: we say that \(T\) is a compatible range (CoR) operator if \(T\) and \(T^\ast\) coincide on \(\mathcal{R}(T) \cap \mathcal{R}(T^\ast)\). We extend and improve some results about DR operators and derive some new results regarding the CoR class.Generalized Drazin invertibility of the product and sum of bounded linear operatorshttps://www.zbmath.org/1483.470032022-05-16T20:40:13.078697Z"Li, Jinfeng"https://www.zbmath.org/authors/?q=ai:li.jinfeng"Wang, Hua"https://www.zbmath.org/authors/?q=ai:wang.hua.1Summary: In this paper, we discuss the existences and representations of the generalized Drazin inverse for the product and sum of two bounded linear operators. Based on the space decomposition method, we prove that the product \(PQ\) and sum \(P+Q\) are generalized Drazin invertible under new conditions, and the expressions of \({ (PQ)^d}\) and \({ (P+Q)^d}\) are given.Drazin invertibility of product and reverse order law of Drazin inverse for bounded linear operatorshttps://www.zbmath.org/1483.470042022-05-16T20:40:13.078697Z"Zhong, Chengcheng"https://www.zbmath.org/authors/?q=ai:zhong.chengcheng"Wang, Hua"https://www.zbmath.org/authors/?q=ai:wang.hua|wang.hua.2|wang.hua.1Summary: In this paper, we discuss the Drazin invertibility of product and the sufficient and necessary conditions of the reverse order laws to hold for the Drazin inverse of bounded linear operators. Firstly, we discuss the Drazin invertibility of the product of three bounded linear operators under commutation conditions \([P, PQ] = 0\) and \([P, Q, R] = 0\), and obtain the sufficient and necessary conditions for the reverse order law of Drazin inverse \({ (PQR)^D} = {R^D}{Q^D}{P^D}\). Secondly, we discuss the Drazin invertibility of the product of two bounded linear operators under commutation conditions \([P, {P^2}Q] = 0\) and \(Q[P, PQ] = 0\), and obtain the sufficient and necessary conditions for the reverse order law of Drazin inverse \({ (PQ)^D} = {Q^D}{P^D}\).On regular linear relationshttps://www.zbmath.org/1483.470052022-05-16T20:40:13.078697Z"Álvarez, T."https://www.zbmath.org/authors/?q=ai:alvarez-seco.teresaSummary: For a closed linear relation in a Banach space, the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semi-Fredholm linear relations which are regular.Polynomial in a Saphar linear relation in a Banach spacehttps://www.zbmath.org/1483.470062022-05-16T20:40:13.078697Z"Álvarez, T."https://www.zbmath.org/authors/?q=ai:alvarez-seco.teresa|alvarez.teresa|alvarez.terezaSummary: In this paper, we introduce the notion of Saphar linear relation in a Banach space and we study the behaviour of such notion in polynomials.Left-right Fredholm and left-right Browder linear relationshttps://www.zbmath.org/1483.470072022-05-16T20:40:13.078697Z"Álvarez, T."https://www.zbmath.org/authors/?q=ai:alvarez-seco.teresa"Fakhfakh, Fatma"https://www.zbmath.org/authors/?q=ai:fakhfakh.fatma"Mnif, Maher"https://www.zbmath.org/authors/?q=ai:mnif.maherSummary: In this paper, we introduce the notions of left (resp., right) Fredholm and left (resp., right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp., right) Browder linear relation \(T\) in a Banach space as an operator-like sum \(T=A+B\), where \(A\) is an injective left (resp., a surjective right) Fredholm linear relation and \(B\) is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp., right) Browder spectrum under finite rank operator.Generalized Kato linear relationshttps://www.zbmath.org/1483.470082022-05-16T20:40:13.078697Z"Benharrat, Mohammed"https://www.zbmath.org/authors/?q=ai:benharrat.mohammed"Álvarez, Teresa"https://www.zbmath.org/authors/?q=ai:alvarez-seco.teresa|alvarez.teresa"Messirdi, Bekkai"https://www.zbmath.org/authors/?q=ai:messirdi.bekkaiSummary: For a Banach space, the notions of generalized Kato linear relation and the corresponding spectrum are introduced and studied. We show that the symmetric difference between the generalized Kato spectrum and the Goldberg spectrum of multivalued linear operators in Banach spaces is at most countable. The obtained results are used to describe the generalized Kato spectrum of the inverse of the left shift operator regarded as a linear relation.Disjoint reiterative \(m_n\)-distributional chaoshttps://www.zbmath.org/1483.470092022-05-16T20:40:13.078697Z"Kostić, Marko"https://www.zbmath.org/authors/?q=ai:kostic.markoSummary: In this paper, we introduce and analyze the notion of disjoint \((m_n,i)\)-distributional chaos, where \(1\leq i\leq 12\), as well as the notions of disjoint \(m_n\)-distributional chaos of type 2 and disjoint reiterative \(m_n\)-distributional chaos of types \(1+\) and \(2^{Bd}\) for general sequences of multivalued linear operators in Fréchet spaces. We reconsider and slightly improve our recent results regarding disjoint distributional chaos in Fréchet spaces.A characterization of the essential approximation pseudospectrum on a Banach spacehttps://www.zbmath.org/1483.470102022-05-16T20:40:13.078697Z"Ammar, Aymen"https://www.zbmath.org/authors/?q=ai:ammar.aymen"Jeribi, Aref"https://www.zbmath.org/authors/?q=ai:jeribi.aref"Mahfoudhi, Kamel"https://www.zbmath.org/authors/?q=ai:mahfoudhi.kamelSummary: One impetus for writing this paper is the issue of approximation pseudospectrum introduced by \textit{M. P. H. Wolff} [J. Approx. Theory 113, No. 2, 229--244 (2001; Zbl 1010.47005)]. The latter study motivates us to investigate the essential approximation pseudospectrum of closed, densely defined linear operators on a Banach space. We begin by defining it and then we focus on the characterization, the stability and some properties of these pseudospectra.Fine structure of the dichotomy spectrumhttps://www.zbmath.org/1483.470112022-05-16T20:40:13.078697Z"Pötzsche, Christian"https://www.zbmath.org/authors/?q=ai:potzsche.christianSummary: The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.Local spectral theory. IIhttps://www.zbmath.org/1483.470122022-05-16T20:40:13.078697Z"Yoo, Jong-Kwang"https://www.zbmath.org/authors/?q=ai:yoo.jong-kwangSummary: In this paper, we show that, if \(A\in L(X)\) and \(B \in L(Y)\), \(X\) and \(Y\) complex Banach spaces, then \(A\oplus B \in L(X \oplus Y)\) is subscalar if and only if both \(A\) and \(B\) are subscalar. We also prove that, if \(A, Q \in L(X)\) satisfies \(AQ = QA\) and \(Q^p = 0\) for some nonnegative integer \(p\), then \(A\) has property \((C)\) (resp., property \((\beta))\) if and only if so does \(A + Q\) (resp., property \((\beta)\)). Finally, we show that \(A\in L(X, Y)\) and \(B, C \in L(Y, X)\) satisfying the operator equation \(ABA = ACA\) and \(BA \in L(X)\) is subscalar with property \((\delta)\), then both \(\operatorname{Lat}(BA)\) and \(\operatorname{Lat}(AC)\) are non-trivial.
For Part I, see [the author, ibid. 38, No. 3--4, 261--269 (2020; Zbl 1463.47022)].Some inequalities for the numerical radius and rhombic numerical radiushttps://www.zbmath.org/1483.470132022-05-16T20:40:13.078697Z"Bajmaeh, Akram Babri"https://www.zbmath.org/authors/?q=ai:bajmaeh.akram-babri"Omidvar, Mohsen Erfanian"https://www.zbmath.org/authors/?q=ai:omidvar.mohsen-erfanianSummary: In this paper, the definition rhombic numerical radius is introduced and we present several numerical radius inequalities. Some applications of these inequalities are considered as well. Particular, it is shown that, if \(A\in\mathcal{B}(\mathcal{H})\) with the Cartesian decomposition \(A=C+iD\) and \(r\geq 1\), then
\[
\begin{aligned}\omega^r(A)\leq\frac{\sqrt{2}}{2}\Vert\vert C+D\vert^{2r}+\vert C-D\vert^{2r}\Vert^{\frac{1}{2}}.\end{aligned}
\]Proper improvement of well-known numerical radius inequalities and their applicationshttps://www.zbmath.org/1483.470142022-05-16T20:40:13.078697Z"Bhunia, Pintu"https://www.zbmath.org/authors/?q=ai:bhunia.pintu"Paul, Kallol"https://www.zbmath.org/authors/?q=ai:paul.kallolThe authors establish some new inequalities for the numerical radius of bounded linear operators. The main section starts with a result that gives an inequality involving the operator norm and the Crawford number of bounded linear operators. Here, new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space \(H\) are mentioned. In particular, it is established that, if \(T\) is a bounded linear operator on a Hilbert space \(H\), then \( w^{2}(T)\leq\min_{ 0\leq\alpha\leq 1} \|\alpha T^*T + (1-\alpha)TT^*\|\), where \(w(T)\) is the numerical radius of \(T\).
The inequalities obtained here are nontrivial improvements of well-known numerical radius inequalities. As an application of the numerical radius inequalities obtained, the authors give a better estimation of the bounds for the zeros of a complex monic polynomial. The paper contains many motivational results.
Reviewer: V. Lokesha (Bangalore)Some refinements of the numerical radius inequalities via Young inequalityhttps://www.zbmath.org/1483.470152022-05-16T20:40:13.078697Z"Heydarbeygi, Z."https://www.zbmath.org/authors/?q=ai:heydarbeygi.zahra"Amyari, M."https://www.zbmath.org/authors/?q=ai:amyari.maryamSummary: In this paper, we get an improvement of the Hölder-McCarthy operator inequality in the case when \(r\geq 1\) and refine generalized inequalities involving powers of the numerical radius for sums and products of Hilbert space operators.Bergman inner functions and \(m\)-hypercontractionshttps://www.zbmath.org/1483.470162022-05-16T20:40:13.078697Z"Eschmeier, Jörg"https://www.zbmath.org/authors/?q=ai:eschmeier.jorgSummary: We characterize operator-valued Bergman inner functions on the unit ball as functions admitting a suitable transfer function realization. Thus we extend corresponding one-variable results of \textit{A. Olofsson} [J. Funct. Anal. 236, No. 2, 517--545 (2006; Zbl 1104.47032); St. Petersbg. Math. J. 19, No. 4, 603--623 (2008; Zbl 1218.47022), translation from Algebra Anal. 19, No. 4, 146--173 (2007)] from the case of the unit disc to the unit ball. At the same time, we associate with each \(m\)-hypercontraction \(T \in L(H)^n\) a~canonical Bergman inner function \(W_T\) and indicate a possible definition of a characteristic function \(\theta_T\) for \(T\).The Helton-Howe trace formula for submoduleshttps://www.zbmath.org/1483.470172022-05-16T20:40:13.078697Z"Fang, Quanlei"https://www.zbmath.org/authors/?q=ai:fang.quanlei"Wang, Yi"https://www.zbmath.org/authors/?q=ai:wang.yi.8|wang.yi.1|wang.yi.4|wang.yi.5|wang.yi.6|wang.yi.7|wang.yi.9|wang.yi.10|wang.yi.3|wang.yi.2"Xia, Jingbo"https://www.zbmath.org/authors/?q=ai:xia.jingboSummary: We consider a class of submodules \(\mathcal{R}\) of the Bergman module \(L_a^2(\mathbf{B})\) that are associated with analytic sets \(\widetilde{M} \subset \mathbf{C}^n\) with \(\dim_{\mathbf{C}} \widetilde{M} = d\). In analogue to the usual Toeplitz operator on \(L_a^2(\mathbf{B})\), we have the ``Toeplitz operator for the submodule'' \( R_\varphi\) on \(\mathcal{R} \). We show that the Helton-Howe trace formula holds for the antisymmetric sum \([ R_{f_1}, R_{f_2}, \dots, R_{f_{2 n}}]\), \(f_1, f_2, \dots, f_{2 n} \in \mathbf{C} [ z_1, \overline{z}_1, \dots, z_n, \overline{z}_n]\).The reducibility of the power of a \(C_0(1)\)-operatorhttps://www.zbmath.org/1483.470182022-05-16T20:40:13.078697Z"Gu, Caixing"https://www.zbmath.org/authors/?q=ai:gu.caixingSummary: Inspired by the work of \textit{R. G. Douglas} and \textit{C. Foiaş} [Oper. Theory: Adv. Appl. 170, 75--84 (2006; Zbl 1119.47010)] on the structure of the square of a \(C_0(1)\)-operator, we form a conjecture about a certain reducibility of any power \(N\) of a \(C_0(1)\)-operator. We then prove this conjecture for \(N = 3\) by determining explicitly the relevant reducing subspaces.Friedrichs extension of operators defined by even order Sturm-Liouville equations on time scaleshttps://www.zbmath.org/1483.470192022-05-16T20:40:13.078697Z"Zemánek, Petr"https://www.zbmath.org/authors/?q=ai:zemanek.petr"Hasil, Petr"https://www.zbmath.org/authors/?q=ai:hasil.petrSummary: In this paper we characterize the Friedrichs extension of operators associated with the 2\(n\)th order Sturm-Liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions. A~nontrivial example is also provided.On the self-regularization of ill-posed problems by the least error projection methodhttps://www.zbmath.org/1483.470202022-05-16T20:40:13.078697Z"Ganina, Alina"https://www.zbmath.org/authors/?q=ai:ganina.alina"Hämarik, Uno"https://www.zbmath.org/authors/?q=ai:hamarik.uno"Kangro, Urve"https://www.zbmath.org/authors/?q=ai:kangro.urveSummary: We consider linear ill-posed problems where both the operator and the right hand side are given approximately. For approximate solution of this equation we use the least error projection method. This method occurs to be a regularization method if the dimension of the projected equation is chosen properly depending on the noise levels of the operator and the right hand side. We formulate the monotone error rule for choice of the dimension of the projected equation and prove the regularization properties.About regularization of severely ill-posed problems by standard Tikhonov's method with the balancing principlehttps://www.zbmath.org/1483.470212022-05-16T20:40:13.078697Z"Solodky, Sergei G."https://www.zbmath.org/authors/?q=ai:solodky.sergei-g"Myleiko, Ganna L."https://www.zbmath.org/authors/?q=ai:myleiko.ganna-lSummary: In the present paper for a stable solution of severely ill-posed problems with perturbed input data, the standard Tikhonov method is applied, and the regularization parameter is chosen according to balancing principle. We establish that the approach provides the order of accuracy \(O((\ln \dots \ln(1/(h+\delta)))^{-p})\) on the class of problems under consideration.Symmetric difference between pseudo B-Fredholm spectrum and spectra originated from Fredholm theoryhttps://www.zbmath.org/1483.470222022-05-16T20:40:13.078697Z"Tajmouati, Abdelaziz"https://www.zbmath.org/authors/?q=ai:tajmouati.abdelaziz"Amouch, Mohamed"https://www.zbmath.org/authors/?q=ai:amouch.mohamed"Karmouni, Mohammed"https://www.zbmath.org/authors/?q=ai:karmouni.mohammedSummary: In this paper, we continue the study of the pseudo B-Fredholm operators of \textit{E. Boasso} [Math. Proc. R. Ir. Acad. 115A, No. 2, 121--135 (2015; Zbl 1345.46041)], and the pseudo B-Weyl spectrum of \textit{H. Zariouh} and \textit{H. Zguitti} [Linear Multilinear Algebra 64, No. 7, 1245--1257 (2016; Zbl 1380.47010)]; in particular, we find that the pseudo B-Weyl spectrum is empty whenever the pseudo B-Fredholm spectrum is, and look at the symmetric differences between the pseudo B-Weyl and other spectra.On the Carey-Helton-Howe-Pincus trace formulahttps://www.zbmath.org/1483.470232022-05-16T20:40:13.078697Z"Chattopadhyay, Arup"https://www.zbmath.org/authors/?q=ai:chattopadhyay.arup"Sinha, Kalyan B."https://www.zbmath.org/authors/?q=ai:sinha.kalyan-bSummary: In this article, we give a new proof of the Carey-Helton-Howe-Pincus trace formula using Kato's theory of ``relatively-smooth'' operators and Krein's trace formula.Herimitian solutions to the equation \(AXA^* + BYB^* = C\) for Hilbert space operatorshttps://www.zbmath.org/1483.470242022-05-16T20:40:13.078697Z"Boussaid, Amina"https://www.zbmath.org/authors/?q=ai:boussaid.amina"Lombarkia, Farida"https://www.zbmath.org/authors/?q=ai:lombarkia.faridaSummary: In this paper, by using generalized inverses we have given some necessary and sufficient conditions for the existence of solutions and Hermitian solutions to some operator equations, and derived a new representation of the general solutions to these operator equations. As a consequence, we have obtained a well-known result of \textit{A. Dajić} and \textit{J. J. Koliha} [J. Math. Anal. Appl. 333, No. 2, 567--576 (2007; Zbl 1120.47009)].Refinements and reverses of Hölder-McCarthy operator inequality via a Cartwright-field resulthttps://www.zbmath.org/1483.470252022-05-16T20:40:13.078697Z"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: By the use of a classical result of \textit{D. I. Cartwright} and \textit{M. J. Field} [Proc. Am. Math. Soc. 71, 36--38 (1978; Zbl 0392.26010)], in this paper, we have obtained new refinements and reverses of Hölder-McCarthy operator inequality in the case of \(p \in (0, 1)\). A~comparison for the two upper bounds obtained showing that neither of them is better in general, has also been performed.Some trace inequalities for operators in Hilbert spaceshttps://www.zbmath.org/1483.470262022-05-16T20:40:13.078697Z"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace inequalities for matrices are also derived. Examples for the operator exponential and other similar functions are presented as well.Extension of the Kantorovich inequality for positive multilinear mappingshttps://www.zbmath.org/1483.470272022-05-16T20:40:13.078697Z"Kian, Mohsen"https://www.zbmath.org/authors/?q=ai:kian.mohsen"Dehghani, Mahdi"https://www.zbmath.org/authors/?q=ai:dehghani.mahdiSummary: It is known that the power function \(f(t)=t^2\) is not matrix monotone. Recently, it has been shown that \(t^2\) preserves the order in some matrix inequalities. We prove that, if \(\mathbb{A} = (A_1,\dots,A_k)\) and \(\mathbb{B} = (B_1,\dots,B_k)\) are \(k\)-tuples of positive matrices with \(0 < m \leq A_i,B_i \leq M\) (\(i=1,\dots,k\)) for some positive real numbers \(m < M\), then
\[
\Phi^2 (A^{-1}_1,\dots,A^{-1}_k) \leq \left(\frac{(1+v^k)^2}{4v^k}\right)^2 \Phi^{-2}(A_1,\dots,A_k)
\]
and
\[
\Phi^2 \left(\frac{A_1+B_1}{2}, \dots,\frac{A_k+B_k}{2}\right) \leq \left(\frac{(1+v^k)^2}{4v^k}\right)^2 \Phi^2 (A_1\sharp B_1, \dots,A_k\sharp B_k),
\]
where \(\Phi\) is a unital positive multilinear mapping and \(v = \frac{M}{m}\) is the condition number of each~\(A_i\).Operator inequalities of Jensen typehttps://www.zbmath.org/1483.470282022-05-16T20:40:13.078697Z"Moslehian, M. S."https://www.zbmath.org/authors/?q=ai:moslehian.mohammad-sal"Mićić, J."https://www.zbmath.org/authors/?q=ai:micic.jadranka"Kian, M."https://www.zbmath.org/authors/?q=ai:kian.mohsenSummary: We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that, if \(f : [0, \infty) \rightarrow \mathbb R\) is a continuous convex function with \(f(0) \leq 0\), then
\[
{\displaystyle{\sum^n_{i=1}}} f(C_i) \leq f \left( {\displaystyle{\sum^n_{i=1}}} C_i \right) - \delta_f {\displaystyle{\sum^n_{i=1}}} \tilde{C}_i \leq f \left( {\displaystyle{\sum^n_{i=1}}} C_i \right)
\]
for all operators \(C_{i}\) such that \(0 \leq C_i \leq M \leq \sum^n_{i=1} C_i\) (\(i = 1, \dots, n\)) for some scalar \(M \geq 0\), where \(\tilde{C}_i = \frac{1}{2} - \left| \frac{C_i}{M} - \frac{1}{2} \right|\) and \(\delta_f = f(0) + f(M) - 2f \left( \frac{M}{2}\right)\).Some Chebyshev type inequalities involving the Hadamard product of Hilbert space operatorshttps://www.zbmath.org/1483.470292022-05-16T20:40:13.078697Z"Teimourian, R."https://www.zbmath.org/authors/?q=ai:teimourian.r"Ghazanfari, A. G."https://www.zbmath.org/authors/?q=ai:ghazanfari.amir-ghasemSummary: In this paper, we prove that, if \(\mathcal{A}\) is a Banach \(*\)-subalgebra of \(B(H)\), \(T\) is a compact Hausdorff space equipped with a Radon measure \(\mu\) and \(\alpha:T\rightarrow [0,\infty)\) is a integrable function and \((A_t)\), \((B_t)\) are appropriate integrable fields of operators in \(\mathcal{A}\) having the almost synchronous property for the Hadamard product, then
\[
\int_T\alpha(s)\,d\mu(s)\int_T\alpha(t)\big(A_t\circ B_t\big) \,d\mu(t) \geq\int_T\alpha(t)A_t\,d\mu(t)\circ\int_T\alpha(t)B_t\,d\mu(t).
\]
We also introduce a semi-inner product for square integrable fields of operators in a Hilbert space and using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product and the trace of operators.Generalized Hellinger metric and Audenaert's in-betweennesshttps://www.zbmath.org/1483.470302022-05-16T20:40:13.078697Z"Dumitru, Raluca"https://www.zbmath.org/authors/?q=ai:dumitru.raluca"Franco, Jose A."https://www.zbmath.org/authors/?q=ai:franco.jose-aSummary: Let \(\sigma\) and \(\tau\) be Kubo-Ando means [\textit{F. Kubo} and \textit{T. Ando}, Math. Ann. 246, 205--224 (1980; Zbl 0412.47013)]. In this article, we consider the in-betweenness property [\textit{K. M. R. Audenaert}, Linear Algebra Appl. 438, No. 4, 1769--1778 (2013; Zbl 1272.47029)] for \(\tau\) with respect to the generalized Hellinger metric induced by \(\sigma\). That is, we show that when \(d_\sigma\) represents the generalized Hellinger metric corresponding to \(\sigma\), \(d_\sigma(A, A \tau B) \leq d_\sigma(A, B) \).The role of algebraic structure in the invariant subspace theoryhttps://www.zbmath.org/1483.470312022-05-16T20:40:13.078697Z"Bagheri-Bardi, G. A."https://www.zbmath.org/authors/?q=ai:bagheri-bardi.ghorban-ali"Elyaspour, A."https://www.zbmath.org/authors/?q=ai:elyaspour.akaram|elyaspour.akram"Esslamzadeh, G. H."https://www.zbmath.org/authors/?q=ai:esslamzadeh.gholam-hosseinSummary: Following a recent work of authors [Linear Algebra Appl. 539, 117--133 (2018; Zbl 06824907)], we establish the algebraic analogues of three major decomposition theorems of Nagy-Foiaş-Langer, Halmos-Wallen and Fishel in Baer \(\ast\)-rings. This approach not only provides purely algebraic and shorter proofs of these decomposition theorems, but also suggests the possibility of eliminating the role of analytic structure in results related to the invariant subspace theory.On the Bari basis properties of the root functions of non-self adjoint \(q\)-Sturm-Liouville problemshttps://www.zbmath.org/1483.470322022-05-16T20:40:13.078697Z"Allahverdiev, B. P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, H."https://www.zbmath.org/authors/?q=ai:tuna.huseyin|tuna.huseinSummary: This paper deals with the dissipative regular \(q\)-Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space \(L_q^2(I)\) by using the asymptotic behavior at infinity for its eigenvalues.Passage of property \((aw)\) from two operators to their tensor producthttps://www.zbmath.org/1483.470332022-05-16T20:40:13.078697Z"Rashid, M. H. M."https://www.zbmath.org/authors/?q=ai:rashid.malik-h-m|rashid.mohammad-hussein-mohammadSummary: A Banach space operator \(S\) satisfies property \((aw)\) if \(\sigma (S)\setminus\sigma_w(S)=E_a^0(S)\), where \(E_a^0(S)\) is the set of all isolated point in the approximate point spectrum which are eigenvalues of finite multiplicity. Property \((aw)\) does not transfer from operators \(A\) and \(B\) to their tensor product \(A\otimes B\), so we give necessary and/or sufficient conditions ensuring the passage of property \((aw)\) from \(A\) and \(B\) to \(A\otimes B\). Perturbations by Riesz operators are considered.Von Neumann type trace inequalities for Schatten-class operatorshttps://www.zbmath.org/1483.470342022-05-16T20:40:13.078697Z"Dirr, Gunther"https://www.zbmath.org/authors/?q=ai:dirr.gunther"vom Ende, Frederik"https://www.zbmath.org/authors/?q=ai:vom-ende.frederikThe von Neumann inequality states that, if \(A,B\in\mathbb{C}^{n\times n}\) with singular values \(s_1(A)\geq s_2(A)\geq\ldots\geq s_n(A)\) and \(s_1(B)\geq s_2(B)\geq\ldots\geq s_n(B)\), respectively, be given, then \[ \max_{U,V \in \mathcal{U}_n}\vert\operatorname{tr}(AUBV)\vert=\sum\nolimits_{j=1}^ns_j (A)s_j(B), \] where \(\mathcal{U}_n\) denotes the unitary group. A~consequence is the von Neumann inequality for Hermitian matrices that reads as follows: Let \(A,B\in \mathbb{C}^{n\times n}\) Hermitian with respective eigenvalues \((\lambda_j(A))_{j=1}^n\) and \((\lambda_j(B))_{j=1}^n\). Then \[ \sum\nolimits_{j=1}^n \lambda_j^\downarrow(A)\lambda_j^\uparrow(B) \leq \operatorname{tr}(AB)\leq \sum\nolimits_{j=1}^n \lambda_j^\downarrow(A)\lambda_j^\downarrow(B), \] where the superindeces \(\downarrow\) and \(\uparrow\) denote the decreasing and increasing sorting of the eigenvectors, respectively. \par The authors of the present paper employ some recent results on the \(C\)-numerical range of Schatten-class operators to extend the above inequalities to Schatten-class operators between complex Hilbert spaces of infinite dimension.
Reviewer: Mohammad Sal Moslehian (Mashhad)More about singular traces on simply generated operator idealshttps://www.zbmath.org/1483.470352022-05-16T20:40:13.078697Z"Pietsch, Albrecht"https://www.zbmath.org/authors/?q=ai:pietsch.albrechtLet \(H\) be a separable Hilbert space. The \(n\)-th approximation number of an operator \(S\) on \(H\) is defined by
\[a_n(S) = \inf\{\|S-F\|: \operatorname{rank}(F) < n\}.\]
A null sequence \((\zeta_h)_h\) is called simple if it is decreasing, with \(\zeta_0=1\), and for some \(c>1\) one has \(\zeta_h \leq c \cdot \zeta_{h+1}\). Then one defines the quasi-Banach operator ideal
\[\mathcal{L}_\infty[\zeta_h](H)=\{S \in \mathcal{L}(H): a_{2^h}(S) = o(\zeta_h)\}\]
with respect to the quasi-norm
\[\|S\|:= \sup_{0\leq h< \infty }\zeta^{-1}_h a_{2^h}(S).\]
The author characterizes certain singular traces linear forms on \(\mathcal{L}_\infty[\zeta_h](H)\) by directly applying Banach's version of the extension theorem
Reviewer: Daniele Puglisi (Catania)Traces of Hilbert space operators and their recent historyhttps://www.zbmath.org/1483.470362022-05-16T20:40:13.078697Z"Pietsch, Albrecht"https://www.zbmath.org/authors/?q=ai:pietsch.albrechtThis paper is a useful systematic survey of the recent theory of traces on operator ideals on Hilbert spaces. Let \(H\) be a Hilbert space and \(\mathcal I(H)\) an operator ideal on \(H\). Recall here that the linear form \(\tau\) on \(\mathcal I(H)\) is a trace on \(\mathcal I(H)\) if \(\tau(BS) = \tau(SB)\) for all \(S \in \mathcal I(X)\) and all bounded operators \(B \in \mathcal L(H)\). Special emphasis in the survey is given to a new approach developed in an extensive series of papers by the author, where traces are associated to certain shift-invariant linear forms on the symmetric sequence space which is related to the given operator ideal \(\mathcal I(H)\), see, e.g., [\textit{A. Pietsch}, Integral Equations Oper. Theory 89, No. 4, 595--606 (2017; Zbl 1464.47016)] and [\textit{A. Pietsch}, Integral Equations Oper. Theory 91, No. 3, Paper No. 21, 29 p. (2019; Zbl 07068609)].
It is also carefully explained how the singular traces fit into this scheme. Recall that the trace \(\tau\) is singular if \(\tau(S) = 0\) for every finite-rank operator \(S\) on \(H\). The first examples of such traces were constructed by \textit{J. Dixmier} [C. R. Acad. Sci., Paris, Sér. A 262, 1107--1108 (1966; Zbl 0141.12902)].
Reviewer: Hans-Olav Tylli (Helsinki)Quasinormality of powers of commuting pairs of bounded operatorshttps://www.zbmath.org/1483.470372022-05-16T20:40:13.078697Z"Curto, Raúl E."https://www.zbmath.org/authors/?q=ai:curto.raul-enrique"Lee, Sang Hoon"https://www.zbmath.org/authors/?q=ai:lee.sanghoon"Yoon, Jasang"https://www.zbmath.org/authors/?q=ai:yoon.jasangSummary: We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal 2-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator \(T\) whose square \(T^2\) is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal \(n\)-tuples. Fourth, we show that if a 2-variable weighted shift \(W_{(\alpha, \beta)}\) and its powers \(W_{(\alpha, \beta)}^{(2, 1)}\) and \(W_{(\alpha, \beta)}^{(1, 2)}\) are all spherically quasinormal, then \(W_{(\alpha, \beta)}\) may not necessarily be jointly quasinormal. Moreover, it is possible for both \(W_{(\alpha, \beta)}^{(2, 1)}\) and \(W_{(\alpha, \beta)}^{(1, 2)}\) to be spherically quasinormal without \(W_{(\alpha, \beta)}\) being spherically quasinormal. Finally, we prove that, for 2-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal.On weighted adjacency operators associated to directed graphshttps://www.zbmath.org/1483.470382022-05-16T20:40:13.078697Z"Exner, George R."https://www.zbmath.org/authors/?q=ai:exner.george-r"Jung, Il Bong"https://www.zbmath.org/authors/?q=ai:jung.il-bong"Lee, Eun Young"https://www.zbmath.org/authors/?q=ai:lee.eun-young"Seo, Minjung"https://www.zbmath.org/authors/?q=ai:seo.minjungSummary: The weighted adjacency operators associated to directed graphs are defined and some operator properties such as normality and hyponormality are investigated. As well, the weighted adjacency operators associated to a forested circuit are defined and their fundamental operator properties are studied. We produce some examples showing the properties of \(p\)-hyponormality and \(p\)-paranormality are distinct for \(p\in(0,\infty)\). Basic connections between directed graphs and Hilbert space operators are discussed.Centered operators via Moore-Penrose inverse and Aluthge transformationshttps://www.zbmath.org/1483.470392022-05-16T20:40:13.078697Z"Jabbarzadeh, M. R."https://www.zbmath.org/authors/?q=ai:jabbarzadeh.mamed-rza-r|jabbarzadeh.mohammd-reza|jabbarzadeh.mohammad-reza"Bakhshkandi, M. Jafari"https://www.zbmath.org/authors/?q=ai:bakhshkandi.m-jafariSummary: In this paper, we obtain some characterizations of centered and binormal operators via Moore-Penrose inverse and Aluthge transform.Polaroid operators and Weyl type theoremshttps://www.zbmath.org/1483.470402022-05-16T20:40:13.078697Z"Mecheri, Salah"https://www.zbmath.org/authors/?q=ai:mecheri.salah"Braha, Naim L."https://www.zbmath.org/authors/?q=ai:braha.naim-latifSubnormal \(n\)th roots of quasinormal operators are quasinormalhttps://www.zbmath.org/1483.470412022-05-16T20:40:13.078697Z"Pietrzycki, Paweł"https://www.zbmath.org/authors/?q=ai:pietrzycki.pawel"Stochel, Jan"https://www.zbmath.org/authors/?q=ai:stochel.janSummary: In [J. Funct. Anal. 278, No. 3, Article ID 108342, 23 p. (2020; Zbl 1483.47037)], \textit{R. E. Curto} et al. asked the following question. \textit{Let A be a subnormal operator, and assume that \( A^2\) is quasinormal. Does it follow that \(A\) is quasinormal?} In this paper, we answer that question in the affirmative. In fact, we prove a more general result that subnormal \(n\)th roots of quasinormal operators are quasinormal. Research on this problem has led us to a new criterion for a semispectral measure on the half-line to be spectral, written in terms of its two ``moments''.Corrigendum to: ``Subnormal \(n\)th roots of quasinormal operators are quasinormal''https://www.zbmath.org/1483.470422022-05-16T20:40:13.078697Z"Pietrzycki, Paweł"https://www.zbmath.org/authors/?q=ai:pietrzycki.pawel"Stochel, Jan"https://www.zbmath.org/authors/?q=ai:stochel.janSummary: In this note, we make a corrigendum of Theorem 4.2 that appeared in our article [ibid. 280, No. 12, Article ID 109001, 14 p. (2021; Zbl 1483.47041)], completing an overlooked fragment and adding the relevant part of the proof.Dilations, models and spectral problems of non-self-adjoint Sturm-Liouville operatorshttps://www.zbmath.org/1483.470432022-05-16T20:40:13.078697Z"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaogluSummary: In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space \(L_{r}^{2}(a,b)\)\ \( (-\infty \leq a<b\leq \infty)\), that [are] the extensions of a minimal symmetric operator\ with defect index (\(2,2\)) (in limit-circle case at singular end points \(a\)\ and \(b\)).\ We examine two classes of dissipative operators with separated boundary conditions and we establish, for each case, a self-adjoint dilation\ of the dissipative operator as well as its incoming and outgoing spectral representations, which enables us to define the scattering matrix of the dilation. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl function of a self-adjoint operator. We present several theorems on completeness of the system of root functions of the dissipative perators and verify them.Non self-adjoint Laplacians on a directed graphhttps://www.zbmath.org/1483.470442022-05-16T20:40:13.078697Z"Balti, Marwa"https://www.zbmath.org/authors/?q=ai:balti.marwaSummary: We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover, we establish isoperimetric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on \textit{heavy end} directed graphs.Norms of composition operators on the \(H^2\) space of Dirichlet serieshttps://www.zbmath.org/1483.470452022-05-16T20:40:13.078697Z"Brevig, Ole Fredrik"https://www.zbmath.org/authors/?q=ai:brevig.ole-fredrik"Perfekt, Karl-Mikael"https://www.zbmath.org/authors/?q=ai:perfekt.karl-mikaelSummary: We consider composition operators \(\mathscr{C}_\varphi\) on the Hardy space of Dirichlet series \(\mathscr{H}^2\), generated by Dirichlet series symbols \(\varphi \). We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of \(\varphi \). The other concerns general symbols, and is based on a geometrical condition on the boundary values of \(\varphi \). Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of \textit{J. H. Shapiro} [Monatsh. Math. 130, No. 1, 57--70 (2000; Zbl 0951.47026)] on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that \(\varphi(s) = c + r 2^{- s} \). A~number of other examples are given.Unbounded composition operators via inductive limits: cosubnormal operators with matrix symbolshttps://www.zbmath.org/1483.470462022-05-16T20:40:13.078697Z"Budzyński, Piotr"https://www.zbmath.org/authors/?q=ai:budzynski.piotr"Dymek, Piotr"https://www.zbmath.org/authors/?q=ai:dymek.piotr"Płaneta, Artur"https://www.zbmath.org/authors/?q=ai:planeta.arturSummary: We prove, by use of inductive techniques, that assorted unbounded composition operators in \(L^2\)-spaces with matrix symbols are cosubnormal.Some essentially normal weighted composition operators on the weighted Bergman spaceshttps://www.zbmath.org/1483.470472022-05-16T20:40:13.078697Z"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-hajiSummary: First of all, we obtain a necessary and sufficient condition for a certain operator \(T_{w}C_{\varphi}\) to be compact on \(A^{2}_{\alpha}\). Next, we give a short proof for Proposition 2.5 which was proved by \textit{B. D. MacCluer} et al. [Complex Var. Elliptic Equ. 58, No. 1, 35--54 (2013; Zbl 1285.47031)]. Then, we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on the weighted Bergman spaces \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism and \(\psi \in H^\infty\) is continuous at a point \(\zeta\) which \(\varphi\) has a finite angular derivative. After that, we find some non-trivially essentially normal weighted composition operators, when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism. In the last section, for \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\), we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(A^{2}_{\alpha}\) and investigate some essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\). Finally, we find some non-trivially essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\).Composition operators in several variables -- a surveyhttps://www.zbmath.org/1483.470482022-05-16T20:40:13.078697Z"Koo, Hyungwoon"https://www.zbmath.org/authors/?q=ai:koo.hyungwoonSummary: In this note we survey progress of composition operators in several variables. We first discuss well-known results in one variables and discuss corresponding progress in several variables. We consider the unit ball, the polydisk in $\mathbb C^n$ as well as $\mathbb C^n$ itself as the underlying domain.
For the entire collection see [Zbl 1261.00015].Differences of generalized weighted composition operators from the Bloch space into Bers-type spaceshttps://www.zbmath.org/1483.470492022-05-16T20:40:13.078697Z"Liu, Xiaosong"https://www.zbmath.org/authors/?q=ai:liu.xiaosong"Li, Songxiao"https://www.zbmath.org/authors/?q=ai:li.songxiaoSummary: We study the boundedness and compactness of the differences of two generalized weighted composition operators acting from the Bloch space to Bers-type spaces.Composition operators between Hardy spaces on linearly convex domains in \(\mathbb{C}^2\)https://www.zbmath.org/1483.470502022-05-16T20:40:13.078697Z"Ly Kim Ha"https://www.zbmath.org/authors/?q=ai:ha.ly-kim"Le Hai Khoi"https://www.zbmath.org/authors/?q=ai:le-hai-khoi.Summary: We study composition operators acting between Hardy spaces \(H^p(\Omega )\), where \(\Omega \subset \mathbb{C}^2\) is a smoothly bounded, \(\mathbb{C} \)-linearly convex domain admitting the so-called \(F\)-type at all boundary points. This \(F\)-type domains contain certain convex domains of finite type and many cases of infinite type in the sense of Range. Criteria for boundedness and compactness of such composition operators are established. Our approach is based on the Cauchy-Leray kernel.Some properties of Moore-Penrose inverse of weighted composition operatorshttps://www.zbmath.org/1483.470512022-05-16T20:40:13.078697Z"Sohrabi, M."https://www.zbmath.org/authors/?q=ai:sohrabi.morteza|sohrabi.mahnaz|sohrabi.mahmoud-reza|sohrabi.maryam|sohrabi.mahmoodSummary: In this paper, we give an explicit formula for the Moore-Penrose inverse of \(W\), denoted by \(W^{\dag}\), on \(L^2(\Sigma)\). As an application, we give a characterization for some operator classes that are weaker than \(p\)-hyponormal with \(W^{\dag}\). Moreover, we give specific examples illustrating these classes.Toeplitz and slant Toeplitz operators on the polydiskhttps://www.zbmath.org/1483.470522022-05-16T20:40:13.078697Z"Hazarika, Munmun"https://www.zbmath.org/authors/?q=ai:hazarika.munmun"Marik, Sougata"https://www.zbmath.org/authors/?q=ai:marik.sougataSummary: For \(n\geq 1\), let \(\mathbb{D}^n\) be the polydisk in \(\mathbb{C}^n\), and let \(\mathbb{T}^n\) be the \(n\)-torus. \(L^2(\mathbb{T}^n)\) denotes the space of Lebesgue square integrable functions on \(\mathbb{T}^n\). In this paper we define slant Toeplitz operators on \(L^2(\mathbb{T}^n)\). Besides giving a necessary and sufficient condition for an operator on \(L^2(\mathbb{T}^n)\) to be slant Toeplitz, we also establish several properties of slant Toeplitz operators.Spectral analysis of certain spherically homogeneous graphshttps://www.zbmath.org/1483.470532022-05-16T20:40:13.078697Z"Breuer, Jonathan"https://www.zbmath.org/authors/?q=ai:breuer.jonathan"Keller, Matthias"https://www.zbmath.org/authors/?q=ai:keller.matthiasSummary: We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.Differences of differential operators between weighted-type spaceshttps://www.zbmath.org/1483.470542022-05-16T20:40:13.078697Z"Al Ghafri, Mohammed Said"https://www.zbmath.org/authors/?q=ai:al-ghafri.mohammed-said"Manhas, Jasbir Singh"https://www.zbmath.org/authors/?q=ai:singh-manhas.jasbirSummary: Let \(\mathcal{H}(\mathbb{D})\) be the space of analytic functions on the unit disc \(\mathbb{D}\). Let \(\psi =(\psi_j)_{j=0}^n\) and \(\Phi =(\Phi_j)_{j=0}^n\) be such that \(\psi_j, \Phi_j \in \mathcal{H}(\mathbb{D})\). The linear differential operator is defined by \(T_{\psi}(f)=\sum_{j=0}^n \psi_j f^{(j)}\), \(f\in \mathcal{H}(\mathbb{D})\). We characterize the boundedness and compactness of the difference operator \((T_{\psi} -T_{\Phi})(f)=\sum_{j=0}^n \left( \psi_j -\Phi_j \right) f^{(j)}\) between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).Operators induced by weighted Toeplitz and weighted Hankel operatorshttps://www.zbmath.org/1483.470552022-05-16T20:40:13.078697Z"Datt, Gopal"https://www.zbmath.org/authors/?q=ai:datt.gopal.1"Mittal, Anshika"https://www.zbmath.org/authors/?q=ai:mittal.anshikaSummary: In this paper, the notion of weighted Toep-Hank operator \(G_{\phi}^{\beta}\), induced by the symbol \(\phi\in L^{\infty}(\beta)\), on the space \(H^2(\beta)\), \(\beta=\{\beta_n\}_{n\in \mathbb{Z}}\) being a semi-dual sequence of positive numbers with \(\beta_0=1\), is introduced. Symbols are identified for the induced weighted Toep-Hank operator to be a co-isometry, normal, and hyponormal.On \(n\)-hynonormality for backward extensions of Bergman weighted shiftshttps://www.zbmath.org/1483.470562022-05-16T20:40:13.078697Z"Dong, Yanwu"https://www.zbmath.org/authors/?q=ai:dong.yanwu"Zheng, Guijun"https://www.zbmath.org/authors/?q=ai:zheng.guijun"Li, Chunji"https://www.zbmath.org/authors/?q=ai:li.chunjiSummary: In this paper, we discuss the backward extensions of Bergman shifts \(W_{\alpha (m)}\), where
\[
{\alpha}(m):\sqrt{\frac{m}{m+1}},{\sqrt{\frac{m+1}{m+2}}},\dots,\ m\in\mathbb{N}.
\]
We obtain a complete description of the \(n\)-hynonormality for backward one, two and three step extensions.Asymptotics of determinants for finite sections of operators with almost periodic diagonalshttps://www.zbmath.org/1483.470572022-05-16T20:40:13.078697Z"Ehrhardt, Torsten"https://www.zbmath.org/authors/?q=ai:ehrhardt.torsten"Zhou, Zheng"https://www.zbmath.org/authors/?q=ai:zhou.zhengSummary: Let \(A = (a_{j, k})_{j, k = - \infty}^\infty\) be a bounded linear operator on \(\ell^2(\mathbb{Z})\) whose diagonals \(D_n(A) = (a_{j, j - n})_{j = - \infty}^\infty \in \ell^\infty(\mathbb{Z})\) are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants \(\det A_{n_1, n_2}\) of the finite sections \(A_{n_1, n_2} = (a_{j, k})_{j, k = n_1}^{n_2 - 1}\) as their size \(n_2 - n_1\) tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.Characterization of linear preservers of generalized majorization on \(c_0\)https://www.zbmath.org/1483.470582022-05-16T20:40:13.078697Z"Eshkaftaki, Ali Bayati"https://www.zbmath.org/authors/?q=ai:eshkaftaki.ali-bayati"Eftekhari, Noha"https://www.zbmath.org/authors/?q=ai:eftekhari.nohaSummary: In this work, we investigate a natural preorder on \(c_0\), the Banach space of all real sequences that tend to zero with the supremum norm, which is said to be ``convex majorization''. Some interesting properties of all bounded linear operators \(T:c_0 \rightarrow c_0\), preserving the convex majorization, are given and we characterize such operators.Generalized Cauchy product and related operators on \(\ell^p(\beta)\)https://www.zbmath.org/1483.470592022-05-16T20:40:13.078697Z"Estaremi, Y."https://www.zbmath.org/authors/?q=ai:estaremi.yousefSummary: In this paper first we give some necessary and sufficient conditions for the boundedness of the multiplication operator \(D_f=M_{*\!\!\!\!\bigcirc,f}\) with respect to the generalized Cauchy product \(*\!\!\!\!\!\bigcirc \), on \(\ell^p(\beta)\). Also, under certain conditions, we give the characterization of the extended eigenvalues and extended eigenvectors of the multiplication operator \(M_{*\!\!\!\!\bigcirc,z}\) on \(\ell^p(\beta)\). Finally we describe the commutants of \(M_{*\!\!\!\!\bigcirc,z}\) and consequently the collection of all hyperinvariant subspaces of \(M_{*\!\!\!\!\bigcirc,z}\).Libera operator on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\) when \(0 < p < 1\)https://www.zbmath.org/1483.470602022-05-16T20:40:13.078697Z"Jevtić, Miroljub"https://www.zbmath.org/authors/?q=ai:jevtic.miroljub"Karapetrović, Boban"https://www.zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: Results from [\textit{M. Pavlović}, ``Definition and properties of the libera operator on mixed norm spaces'', Sci. World J. 2014, Article ID 590656, 15 p. (2014; \url{doi:10.1155/2014/590656})] on Libera operator acting on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\), for \(1 \leq p \leq \infty\), are extended to the case \(0 < p < 1\).Fine spectra of upper triangular triple-band matrices over the sequence space \(\ell_p\) (\(0 < p < \infty\))https://www.zbmath.org/1483.470612022-05-16T20:40:13.078697Z"Karaisa, Ali"https://www.zbmath.org/authors/?q=ai:karaisa.ali"Başar, Feyzi"https://www.zbmath.org/authors/?q=ai:basar.feyziSummary: The fine spectra of lower triangular triple-band matrices have been examined by several authors, e.g., [\textit{A. M. Akhmedov} and the second author, Demonstr. Math. 39, No. 3, 585--595 (2006; Zbl 1118.47303); Acta Math. Sin., Engl. Ser. 23, No. 10, 1757--1768 (2007; Zbl 1134.47025); \textit{H. Furkan} et al., Comput. Math. Appl. 60, No. 7, 2141--2152 (2010; Zbl 1222.47050)]. Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space \(\ell_p\). The operator \(A(r, s, t)\) on sequence space on \(\ell_p\) is defined by \(A(r, s, t)x = (rx_k + sx_{k + 1} + tx_{k + 2})^\infty_{k = 0}\), where \(x = (x_k) \in \ell_p\), with \(0 < p < \infty\). In this paper we have obtained the results on the spectrum and point spectrum for the operator \(A(r, s, t)\) on the sequence space \(\ell_p\). Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator \(A(r, s, t)\) on the sequence space \(\ell_p\) are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator \(A(r, s, t)\) over the space \(\ell_p\) and we give some applications.Existence of non-subnormal completely semi-weakly hyponormal weighted shiftshttps://www.zbmath.org/1483.470622022-05-16T20:40:13.078697Z"Li, Chunji"https://www.zbmath.org/authors/?q=ai:li.chunji"Lee, Mi Ryeong"https://www.zbmath.org/authors/?q=ai:lee.mi-ryeongSummary: In this paper, we introduce a new notion of completely semi-weakly hyponormal operator which is a special case of polynomially hyponormal operator. For an one-step backward extension of the Bergman weighted shift, we show that completely semi-weakly hyponormal weighted shifts need not be subnormal. In addition, we provide an example which can serve to distinguish the semi-weak \(m\)-hyponormality from the semi-weak \(m\)-hyponormality with positive determinant coefficients for such a shift. Finally, we discuss flatness on semi-weakly \(m\)-hyponormal weighted shifts.\(C\)-selfadjointness of the product of a composition operator and a maximal differentiation operatorhttps://www.zbmath.org/1483.470632022-05-16T20:40:13.078697Z"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-haji"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Hai, Pham Viet"https://www.zbmath.org/authors/?q=ai:pham-viet-hai.Summary: Let \(\varphi\) be an automorphism of \(\mathbb{D}\). In this paper, we consider the operator \(C_\varphi D_{\psi_0,\psi_1}\) on the Hardy space \(H^2\) which is the product of composition and the maximal differential operator. We characterize these operators which are \(C\)-selfadjoint with respect to some conjugations \(C\). Moreover, we find all Hermitian operators \(C_\varphi D_{\psi_0,\psi_1}\), when \(\varphi\) is a rotation.Norm of some operators from logarithmic Bloch-type spaces to weighted-type spaceshttps://www.zbmath.org/1483.470642022-05-16T20:40:13.078697Z"Stević, Stevo"https://www.zbmath.org/authors/?q=ai:stevic.stevoSummary: Operator norm of weighted composition operators from the iterated logarithmic Bloch space \(\mathcal{B}_{{\log}_k} , k\in \mathbb{N}\), or the logarithmic Bloch-type space \(\mathcal{B}_{{\log}^{\beta}},\beta \in (0,1)\) to weighted-type spaces on the unit ball are calculated. It is also calculated norm of the product of differentiation and composition operators among these spaces on the unit disk.Generalized Cesàro operators, fractional finite differences and gamma functionshttps://www.zbmath.org/1483.470652022-05-16T20:40:13.078697Z"Abadias, Luciano"https://www.zbmath.org/authors/?q=ai:abadias.luciano"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-jSummary: In this paper, we present a complete spectral research of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable \(C_0\)-semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators.Generalized Volterra operators on polynomially generated Banach spaceshttps://www.zbmath.org/1483.470662022-05-16T20:40:13.078697Z"Eghbali, Nasrin"https://www.zbmath.org/authors/?q=ai:eghbali.nasrin"Pirasteh, Maryam M."https://www.zbmath.org/authors/?q=ai:pirasteh.maryam-m"Sanatpour, Amir H."https://www.zbmath.org/authors/?q=ai:sanatpour.amir-hosseinSummary: We study boundedness of generalized Volterra operators acting on certain Banach spaces of analytic functions generated by the polynomials on the open unit disc. The operators under study, map into the weighted Banach spaces of analytic functions or Bloch type spaces. We also give some related results for the boundedness of continuous operators with respect to the topology of uniform convergence on compact subsets of the open unit disc.Additive results for Moore-Penrose inverse of Lambert conditional operatorshttps://www.zbmath.org/1483.470672022-05-16T20:40:13.078697Z"Sohrabi, M."https://www.zbmath.org/authors/?q=ai:sohrabi.morteza|sohrabi.mahnaz|sohrabi.mahmoud-reza|sohrabi.mahmood|sohrabi.maryamLet \((X,\Sigma, \mu)\) be a complete \(\Sigma\)-finite measure space. We denote the linear space of all complex-valued \(\Sigma\)-measurable functions on \(X\) by \(L^{0}(\Sigma)\). For \(u,w\in L^{0}(\Sigma)\), let \(T:L^{2}(\Sigma)\to L^{2}(\Sigma)\) be \[T(f):=wE(uf),\] where \(E\) means the conditional expectation. It is called the Lambert multiplication operator. \(T\) is represented by \(T=M_{w}EM_{u}\), where \(M_{u}\) means a multiplication operator, i.e., \(M_{u}(f):=uf\) for \(f\in L^{2}(\Sigma)\).\par In this paper, the author gives several properties of \(T\) and its Moore-Penrose inverse \(T^{\dagger}\). The author obtains equivalent conditions of \(T\) and \(T^{\dagger}\) for normality and binormality. Moreover, the author obtains an equivalent condition that \((T_{1}T_{2}T_{3})^{\dagger}=T_{3}^{\dagger}T_{2}^{\dagger}T_{1}^{\dagger}\) holds for Lambert multiplication operators \(T_{1}, T_{2}\) and \(T_{3}\). Lastly, the author considers upper and lower estimations of the numerical radii of \(T\) and \(T^{\dagger}\).
Reviewer: Takeaki Yamazaki (Kawagoe)Some conditions under which left derivations are zerohttps://www.zbmath.org/1483.470682022-05-16T20:40:13.078697Z"Hosseini, Amin"https://www.zbmath.org/authors/?q=ai:hosseini.aminSummary: In this study, we show that every continuous Jordan left derivation on a (commutative or noncommutative) prime UMV-Banach algebra with the identity element~1 is identically zero. Moreover, we prove that every continuous left derivation on a unital finite dimensional Banach algebra, under certain conditions, is identically zero. As another result in this regard, it is proved that if \(\mathfrak{R}\) is a 2-torsion free semiprime ring such that [the annihilator] \(\operatorname{ann}\{[y,z]\mid y,z \in \mathfrak{R}\}=\{0\}\), then every Jordan left derivation \(\mathfrak{L}:\mathfrak{R}\rightarrow \mathfrak{R}\) is identically zero. In addition, we provide several other results in this regard.On the invariance of primitive ideals via \({\phi}\)-derivations on Banach algebrashttps://www.zbmath.org/1483.470692022-05-16T20:40:13.078697Z"Jung, Yong-Soo"https://www.zbmath.org/authors/?q=ai:jung.yong-sooSummary: The noncommutative Singer-Wermer conjecture [\textit{I. M. Singer} and \textit{J. Wermer}, Math. Ann. 129, 260--264 (1955; Zbl 0067.35101)] states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of \(\phi\)-derivations to be nilpotent. Moreover, we show that the spectral boundedness of \(\phi\)-derivations implies that they leave each primitive ideal of Banach algebras invariant.Asymetric Fuglede Putnam's theorem for operators reduced by their eigenspaceshttps://www.zbmath.org/1483.470702022-05-16T20:40:13.078697Z"Lombarkia, Farida"https://www.zbmath.org/authors/?q=ai:lombarkia.farida"Amouch, Mohamed"https://www.zbmath.org/authors/?q=ai:amouch.mohamedSummary: Fuglede-Putnam theorem has been proved for a considerably large number of class of operators. In this paper, by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.Derivations on FCIN algebrashttps://www.zbmath.org/1483.470712022-05-16T20:40:13.078697Z"Majeed, Asia"https://www.zbmath.org/authors/?q=ai:majeed.asia"Özel, Cenap"https://www.zbmath.org/authors/?q=ai:ozel.cenapSummary: Let \(\mathcal{L}\) be an algebra generated by the commuting independent nests, \(\mathcal{M}\) is an ultra-weakly closed subalgebra of \(\mathbf{B}(\mathbf{H})\) which contains \(\operatorname{alg}\mathcal{L}\) and \(\phi\) is a norm continuous linear mapping from \(\operatorname{alg}\mathcal{L}\) into \(\mathcal{M}\). In this paper, we will show that a norm continuous linear derivable mapping at zero point from \(\operatorname{Alg}\mathcal{L}\) to \(\mathcal{M}\) is a derivation.Characterizing Jordan homomorphismshttps://www.zbmath.org/1483.470722022-05-16T20:40:13.078697Z"Mathieu, Martin"https://www.zbmath.org/authors/?q=ai:mathieu.martinSummary: It is shown that every bounded, unital linear mapping that preserves elements of square zero from a \(C^*\)-algebra of real rank zero and without tracial states into a Banach algebra is a Jordan homomorphism.When is a bi-Jordan homomorphism bi-homomorphism?https://www.zbmath.org/1483.470732022-05-16T20:40:13.078697Z"Zivari-Kazempour, A."https://www.zbmath.org/authors/?q=ai:zivari-kazempour.abbasSummary: For Banach algebras \(\mathcal{A}\) and \(\mathcal{B}\), we show that, if \(\mathcal{U}=\mathcal{A}\times\mathcal{B}\) is commutative (weakly commutative), then each bi-Jordan homomorphism from \(\mathcal{U}\) into a semisimple commutative Banach algebra \(\mathcal{D}\) is a bi-homomorphism. We also prove the same result for 3-bi-Jordan homomorphism with the additional hypothesis that the Banach algebra \(\mathcal{U}\) is unital.Norm inequalities for elementary operators and other inner product type integral transformers with the spectra contained in the unit dischttps://www.zbmath.org/1483.470742022-05-16T20:40:13.078697Z"Jocić, Danko R."https://www.zbmath.org/authors/?q=ai:jocic.danko-r"Milošević, Stefan"https://www.zbmath.org/authors/?q=ai:milosevic.stefan"Đurić, Vladimir"https://www.zbmath.org/authors/?q=ai:duric.vladimirSummary: If \(\{\mathscr{A}_t\}_{t\in\Omega}\) and \(\{\mathscr{B}_t\}_{t\in\Omega}\) are weakly*-measurable families of bounded Hilbert space operators such that transformers \(X \mapsto \int_\Omega \mathscr{A}_t^\ast X\mathscr{A}_t \,d\mu(t)\) and \(X \mapsto \int_\Omega \mathscr{B}_t^\ast X \mathscr{B}_t \,d\mu(t)\) on \(\mathcal{B}(\mathcal{H})\) have their spectra contained in the unit disc, then for all bounded operators \(X\)
\[
\|\Delta_{\mathscr{A}} X \Delta_{\mathscr{B}}\| \leqslant \left\|X - \int_\Omega \mathscr{A}_t^\ast X \mathscr{B}_t d\mu(t)\right\|,
\tag{1}
\]
where \(\Delta_{\mathscr{A}} := s-\lim_{r\nearrow 1}\left(I + \sum_{n+1}^\infty r^{2n} \int_\Omega \big| \mathscr{A}_{t_1} \dots \mathscr{A}_{t_n} \big|^2 d\mu^n(t_1,\dots,t_n)\right)^{-1/2}\) and \(\Delta_{\mathscr{B}}\) by analogy.
If, additionally, \(\sum_{n=1}^\infty \int_{\Omega^n} \big| \mathscr{A}_{t_1}^\ast \dots\mathscr{A}_{t_n}^\ast \big|^2 d\mu^n(t_1,\dots,t_n)\) and \(\sum_{n=1}^\infty \int_{\Omega^n} \big| \mathscr{B}_{t_1}^\ast \dots \mathscr{B}_{t_n}^\ast \big|^2 d\mu^n (t_1,\dots,t_n)\) both represent bounded operators, then for all \(p,q,s \geqslant 1\) such that \(\frac{1}{q}+\frac{1}{s}=\frac{2}{p}\) and for all Schatten \(p\) trace class operators \(X\)
\[
\left\|\Delta_{\mathscr{A}}^{1-\frac{1}{q}} X\Delta_{\mathscr{B}}^{1-\frac{1}{s}}\right\|_p \leqslant \left\|\Delta_{\mathscr{A}^\ast}^{-\frac{1}{q}} \Big(X - \int_\Omega \mathscr{A}_t^\ast X \mathscr{B}_t \,d\mu(t)\Big)\Delta_{\mathscr{B}^\ast}^{-\frac{1}{s}}\right\|_p.
\tag{2}
\]
If at least one of those families consists of bounded commuting normal operators, then (1) holds for all unitarily invariant Q-norms. Applications to shift operators are also given.Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operatorshttps://www.zbmath.org/1483.470752022-05-16T20:40:13.078697Z"Pankov, Mark"https://www.zbmath.org/authors/?q=ai:pankov.markSummary: Let \(H\) be a complex Hilbert space and let \(\mathcal{C}\) be a conjugacy class of rank \(k\) self-adjoint operators on \(H\) with respect to the action of the group of unitary operators. Under the assumption that \(\dim H \geq 4 k\), we describe all bijective transformations of \(\mathcal{C}\) preserving the commutativity in both directions. In particular, it follows from this description that every such transformation is induced by a unitary or anti-unitary operator only in the case when for every operator from \(\mathcal{C}\) the dimensions of eigenspaces are mutually distinct.AM-totally boundedness on normed Riesz spaceshttps://www.zbmath.org/1483.470762022-05-16T20:40:13.078697Z"Baklouti, H."https://www.zbmath.org/authors/?q=ai:baklouti.hamadi"Hajji, M."https://www.zbmath.org/authors/?q=ai:hajji.mansour|hajji.mohamed-karim|hajji.mohamed-ali"Moulahi, R."https://www.zbmath.org/authors/?q=ai:moulahi.radhoueneIn this paper, the authors introduce the class of AM-totally bounded operators acting on vector lattices and they use the new norm to study the domination problem in this operator class. They give some new results concerning the domination power problem by totally bounded operators acting on vector lattices.
Reviewer: Ömer Gök (İstanbul)BMO-estimates for non-commutative vector valued Lipschitz functionshttps://www.zbmath.org/1483.470772022-05-16T20:40:13.078697Z"Caspers, M."https://www.zbmath.org/authors/?q=ai:caspers.martijn"Junge, M."https://www.zbmath.org/authors/?q=ai:junge.marius"Sukochev, F."https://www.zbmath.org/authors/?q=ai:sukochev.fedor-a"Zanin, D."https://www.zbmath.org/authors/?q=ai:zanin.dmitriy-vSummary: We construct Markov semigroups \(\mathcal{T}\) and associated BMO-spaces on a finite von Neumann algebra \((\mathcal{M}, \tau)\) and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that, for any \(A \in \mathcal{M}\) self-adjoint and \(f : \mathbb{R} \rightarrow \mathbb{R}\) Lipschitz, there is a Markov semigroup \(\mathcal{T}\) such that, for \(x \in \mathcal{M}\), \[\| [f(A), x] \|_{\mathrm{bmo}(\mathcal{M}, \mathcal{T})} \leq c_{a b s} \| f^\prime \|_\infty \| [A, x] \|_\infty .\] We obtain an analogue of this result for more general von Neumann valued-functions \(f : \mathbb{R}^n \rightarrow \mathcal{N}\) by imposing Hörmander-Mikhlin type assumptions on \(f\).
In establishing these results, we show that Markov dilations of Markov semigroups have certain automatic continuity properties. We also show that Markov semigroups of double operator integrals admit (standard and reversed) Markov dilations.On the Wold-type decompositions for \(n\)-tuples of commuting isometric semigroupshttps://www.zbmath.org/1483.470782022-05-16T20:40:13.078697Z"Bînzar, Tudor"https://www.zbmath.org/authors/?q=ai:binzar.tudor"Lăzureanu, Cristian"https://www.zbmath.org/authors/?q=ai:lazureanu.cristianSummary: In this paper, the \(n\)-tuples of commuting isometric semigroups on a Hilbert space and the product semigroup generated by them are considered. Properties of the right defect spaces and characterizations of the semigroups of type ``s'' are given. Also, the Wold-type decompositions with \(3^n\) summands for \(n\)-tuples of commuting isometric semigroups are introduced. The existence and uniqueness of such decompositions are analysed and several connections with the Wold decompositions of each semigroup and their product semigroup are presented.On higher regularized traces of a differential operator with bounded operator coefficient given in a finite intervalhttps://www.zbmath.org/1483.470792022-05-16T20:40:13.078697Z"Sezer, Yonca"https://www.zbmath.org/authors/?q=ai:sezer.yonca"Bakşi, Özlem"https://www.zbmath.org/authors/?q=ai:baksi.ozlem"Karayel, Serpil"https://www.zbmath.org/authors/?q=ai:karayel.serpil-sengulSummary: In this work, we find a higher regularized trace formula for a regular Sturm-Liouville differential operator with operator coefficient.Quasispectral decomposition of the heat potentialhttps://www.zbmath.org/1483.470802022-05-16T20:40:13.078697Z"Kal'menov, T. Sh."https://www.zbmath.org/authors/?q=ai:kalmenov.tynysbek-sharipovich"Arepova, G. D."https://www.zbmath.org/authors/?q=ai:arepova.gaukhar-dSummary: In this paper, by multiplying with the unitary operator \((Pf)(x,t)= f(x,T-t)\), \(0\le t\le T\), the \(\lozenge^{-1}\) heat potential turns into a self-adjoint operator \(P\lozenge^{-1}\). From the spectral decomposition of a completely continuous self-adjoint operator \(P\lozenge^{-1}\) we obtain the quasi spectral decomposition of the heat potential operator.Variational inclusion governed by \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappinghttps://www.zbmath.org/1483.470812022-05-16T20:40:13.078697Z"Gupta, Sanjeev"https://www.zbmath.org/authors/?q=ai:gupta.sanjeev-k"Husain, Shamshad"https://www.zbmath.org/authors/?q=ai:husain.shamshad"Mishra, Vishnu Narayan"https://www.zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we look into a new concept of accretive mappings called \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings connected with generalized \(m\)-accretive mappings to the \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings and discuss its characteristics like single-valuable [sic] and Lipschitz continuity. Some illustration are given in support of \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings. Since proximal point mapping is a powerful tool for solving variational inclusion. Therefore, as an application of introduced mapping, we construct an iterative algorithm to solve variational inclusions and show its convergence with acceptable assumptions.PPF dependent fixed points of generalized contractive type mappings using \(C\)-class functions with an applicationhttps://www.zbmath.org/1483.470822022-05-16T20:40:13.078697Z"Babu, Gutti Venkata Ravindranadh"https://www.zbmath.org/authors/?q=ai:babu.gutti-venkata-ravindranadh"Kumar, Madugula Vinod"https://www.zbmath.org/authors/?q=ai:kumar.madugula-vinodSummary: In this paper, we prove the existence of PPF dependent fixed points of single-valued generalized \(\alpha-\eta-\psi-\phi-F-\)contraction type mappings and extend it to multi-valued \(\alpha^*-\psi-\phi-F-\)contraction type mappings in Banach spaces. Also, we introduce the concept of \(f-\alpha^*-\)admissible mapping and prove the existence of PPF dependent coincidence points of a pair of single-valued and multi-valued mappings. A~fixed point result in a Banach space endowed with a graph is obtained as an application of PPF dependent fixed point result of a single-valued mapping.\(\rho\)-attractive elements in modular function spaceshttps://www.zbmath.org/1483.470832022-05-16T20:40:13.078697Z"Iqbal, H."https://www.zbmath.org/authors/?q=ai:iqbal.hira"Abbas, M."https://www.zbmath.org/authors/?q=ai:abbas.mujahid"Khan, S. H."https://www.zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: In this paper, we introduce the notion of \(\rho\)-attractive elements in modular function spaces. A new class of mappings called \(\rho\)-\(k\)-nonspreading mappings is also introduced. Making a good use of the two notions, we first prove existence results and then some approximation results in the setup of modular function spaces. An example is presented to support the results proved herein.Existence and structure of the common fixed points based on TVShttps://www.zbmath.org/1483.470842022-05-16T20:40:13.078697Z"Mohamadi, Issa"https://www.zbmath.org/authors/?q=ai:mohamadi.issa"Saeidi, Shahram"https://www.zbmath.org/authors/?q=ai:saeidi.shahramSummary: In this paper, we investigate the common fixed point property for commutative nonexpansive mappings on \(\tau\)-compact convex sets in normed and Banach spaces, where \(\tau\) is a Hausdorff topological vector space topology that is weaker than the norm topology. As a consequence of our main results, we obtain that the set of common fixed points of any commutative family of nonexpansive self-mappings of a nonempty \(clm\)-compact (resp., weak* compact) convex subset \(C\) of \(L_1(\mu)\) with a \(\sigma\)-finite \(\mu\) (resp., the James space \(J_0\)) is a nonempty nonexpansive retract of \(C\).A nonlinear \(F\)-contraction form of Sadovskii's fixed point theorem and its application to a functional integral equation of Volterra typehttps://www.zbmath.org/1483.470852022-05-16T20:40:13.078697Z"Nourouzi, Kourosh"https://www.zbmath.org/authors/?q=ai:nourouzi.kourosh"Zahedi, Faezeh"https://www.zbmath.org/authors/?q=ai:zahedi.faezeh"O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we give a nonlinear \(F\)-contraction form of the Sadovskii fixed point theorem and we also investigate the existence of solutions for a functional integral equation of Volterra type.A new class of contractive mappingshttps://www.zbmath.org/1483.470862022-05-16T20:40:13.078697Z"Popescu, O."https://www.zbmath.org/authors/?q=ai:popescu.otilia|popescu.ovidiu|popescu.octavThe author introduces a new class of mappings including enriched contractions, enriched Kannan mappings and enriched Chatterjea mappings, and then he proves some fixed point theorems for these mappings. Some examples illustrating the obtained results are finally presented.
Reviewer: Rodica Luca (Iaşi)Fixed points of mappings over a locally convex topological vector space and Ulam-Hyers stability of fixed point problemshttps://www.zbmath.org/1483.470872022-05-16T20:40:13.078697Z"Roy, Kushal"https://www.zbmath.org/authors/?q=ai:roy.kushal"Saha, Mantu"https://www.zbmath.org/authors/?q=ai:saha.mantuSummary: This paper is dealt with the theory of fixed points of mappings which are an analogue like contraction mapping and Kannan mappings over a locally convex topological vector space. Some common fixed point theorems for a pair of mappings involving their iterates have been proved. The purpose of this paper is to examine the validity of established results on fixed points of contraction mappings and Kannan mappings over such a locally convex topological vector space. It is revealed that a suitable local base in locally convex topological vector space plays an important role to find fixed points of above mappings over it. Also an application relating to stability of fixed point equation for Kannan-type contractive mappings is obtained here.Forward-backward approximation of nonlinear semigroups in finite and infinite horizonhttps://www.zbmath.org/1483.470882022-05-16T20:40:13.078697Z"Contreras, Andrés"https://www.zbmath.org/authors/?q=ai:contreras.andres-a"Peypouquet, Juan"https://www.zbmath.org/authors/?q=ai:peypouquet.juanThe authors consider the problem
\[
\begin{aligned}
-&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\
&u(0)=u_{0}\in D(A),
\end{aligned}
\]
in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].
Reviewer: Daniel C. Biles (Nashville)A mean ergodic theorem for nonexpansive mappings in Hadamard spaceshttps://www.zbmath.org/1483.470892022-05-16T20:40:13.078697Z"Khatibzadeh, H."https://www.zbmath.org/authors/?q=ai:khatibzadeh.hadi"Pouladi, H."https://www.zbmath.org/authors/?q=ai:pouladi.hadiThe authors generalize the Baillon nonlinear ergodic theorem to the case of nonexpansive mappings in Hadamard (nonpositive curvature complete metric) spaces. Namely, they shows that the Karcher means of the iterations converges weakly to a fixed point of the mapping, which is supposed to exist. This result is also extended for \(1\)-parameter continuous semigroups of contractions.
Reviewer: Evgeniy Panov (Veliky Novgorod)Nonlinear ergodic theorem for commutative families of positively homogeneous nonexpansive mappings in Banach spaces and applicationshttps://www.zbmath.org/1483.470902022-05-16T20:40:13.078697Z"Takahashi, Wataru"https://www.zbmath.org/authors/?q=ai:takahashi.wataru"Wong, Ngai-Ching"https://www.zbmath.org/authors/?q=ai:wong.ngai-ching"Yao, Jen-Chih"https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: Recently, two retractions (projections) which are different from the metric projection and the sunny nonexpansive retraction in a Banach space were introduced. In this paper, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for a commutative family of positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions. In the proof, we use the theory of invariant means essentially. We apply our nonlinear ergodic theorem to get some nonlinear ergodic theorems in Banach spaces.Strict feasibility of variational inclusion problems in reflexive Banach spaceshttps://www.zbmath.org/1483.470912022-05-16T20:40:13.078697Z"Luo, Xue-Ping"https://www.zbmath.org/authors/?q=ai:luo.xueping"Xiao, Yi-Bin"https://www.zbmath.org/authors/?q=ai:xiao.yibin"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11Summary: In this paper, we are devoted to introducing the strict feasibility of a variational inclusion problem as a novel notion. After proving a new equivalent characterization for the nonemptiness and boundedness of the solution set for the variational inclusion problem under consideration, it is proved that the nonemptiness and boundedness of the solution set for the variational inclusion problem with a maximal monotone mapping is equivalent to its strict feasibility in reflexive Banach spaces.Relaxed \(\eta\)-proximal operator for solving a variational-like inclusion problemhttps://www.zbmath.org/1483.470922022-05-16T20:40:13.078697Z"Rahaman, Mijanur"https://www.zbmath.org/authors/?q=ai:rahaman.mijanur"Ahmad, Rais"https://www.zbmath.org/authors/?q=ai:ahmad.rais"Dilshad, Mohd"https://www.zbmath.org/authors/?q=ai:dilshad.mohd"Ahmad, Iqbal"https://www.zbmath.org/authors/?q=ai:ahmad.iqbalSummary: In this paper, we introduce a new resolvent operator and we call it relaxed \(\eta\)-proximal operator. We demonstrate some of the properties of relaxed \(\eta\)-proximal operator. By applying this concept, we consider and study a variational-like inclusion problem with a nonconvex functional. Based on relaxed \(\eta\)-proximal operator, we define an iterative algorithm to approximate the solutions of a variational-like inclusion problem and the convergence of the iterative sequences generated by the algorithm is also discussed. Our results can be treated as refinement of many previously known results. An example is constructed in support of Theorem 1.An iterative method for solution of finite families of split minimization problems and fixed point problemshttps://www.zbmath.org/1483.470932022-05-16T20:40:13.078697Z"Abass, Hammed Anuoluwapo"https://www.zbmath.org/authors/?q=ai:abass.hammed-anuoluwapo"Izuchukwu, Chinedu"https://www.zbmath.org/authors/?q=ai:izuchukwu.chinedu"Mewomo, Oluwatosin Temitope"https://www.zbmath.org/authors/?q=ai:mewomo.oluwatosin-temitope"Ogbuisi, Ferdinard Udochukwu"https://www.zbmath.org/authors/?q=ai:ogbuisi.ferdinard-udochukwuSummary: The purpose of this paper is to introduce a proximal iterative algorithm for the approximation of a common solution of finite families of split minimization problem and a fixed point problem in the framework of Hilbert space. Using our iterative algorithm, we prove a strong convergence theorem for approximating a common solution of finite families of split minimization problem and a fixed point problem of nonexpansive mapping. Moreover, our result complements and extends some related results in literature.Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaceshttps://www.zbmath.org/1483.470942022-05-16T20:40:13.078697Z"Auwalu, Abba"https://www.zbmath.org/authors/?q=ai:auwalu.abba"Mohammed, Lawan Bulama"https://www.zbmath.org/authors/?q=ai:mohammed.lawan-bulama"Saliu, Afis"https://www.zbmath.org/authors/?q=ai:saliu.afisSummary: In this paper, we study synchronal and cyclic algorithms for finding a common fixed point \(x^*\) of a finite family of strictly pseudocontractive mappings, which solve the variational inequality
\[
\langle (\gamma f - \mu G)x^*, j_q(x-x^*)\rangle \leq 0, \quad \forall x \in \cap_{i=1}^N F(T_i),
\]
where \(f\) is a contraction mapping, \(G\) is an \(\eta\)-strongly accretive and \(L\)-Lipschitzian operator, \(N \geq 1\) is a positive integer, \(\gamma, \mu >0\) are arbitrary fixed constants, and \(\{T_i\}_{i=1}^N\) are \(N\)-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real \(q\)-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.Proximal point algorithm for a countable family of weighted resolvent averageshttps://www.zbmath.org/1483.470952022-05-16T20:40:13.078697Z"Bagheri, Malihe"https://www.zbmath.org/authors/?q=ai:bagheri.malihe"Roohi, Mehdi"https://www.zbmath.org/authors/?q=ai:roohi.mehdiSummary: In this paper, we introduce a composite iterative algorithm for finding a common zero point of a countable family of weighted resolvent average of finite family of monotone operators in Hilbert spaces. We prove that the sequence generated by the iterative algorithm converges strongly to a common zero point. Finally, we apply our results to split common zero point problem.On strong convergence of Halpern's method for quasi-nonexpansive mappings in Hilbert spaceshttps://www.zbmath.org/1483.470962022-05-16T20:40:13.078697Z"Falset, Jesús Garcia"https://www.zbmath.org/authors/?q=ai:garcia-falset.jesus"Llorens-Fuster, Enrique"https://www.zbmath.org/authors/?q=ai:llorens-fuster.enrique"Marino, Giuseppe"https://www.zbmath.org/authors/?q=ai:marino.giuseppe"Rugiano, Angela"https://www.zbmath.org/authors/?q=ai:rugiano.angelaSummary: In this paper, we introduce a Halpern's type method to approximate common fixed points of a nonexpansive mapping \(T\) and a strongly quasi-nonexpansive mappings \(S\), defined in a Hilbert space, such that \(I - S\) is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given.A novel low-cost method for generalized split inverse problem of finite family of demimetric mappingshttps://www.zbmath.org/1483.470972022-05-16T20:40:13.078697Z"Gebrie, Anteneh Getachew"https://www.zbmath.org/authors/?q=ai:gebrie.anteneh-getachewSummary: The purpose of this paper is to introduce iterative method for solving generalized split inverse problem given as a task of finding a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. The proposed algorithm is formulated in low-cost sequential computing method with step size selection technique. The strong convergence theorem of the proposed algorithm is derived under the appropriate assumptions. The result presented in the paper generalizes several results in the literature. Numerical example is given to illustrate the efficiency and performance of the proposed iterative method.On a variable metric iterative method for solving the elastic net problemhttps://www.zbmath.org/1483.470982022-05-16T20:40:13.078697Z"Liu, Liya"https://www.zbmath.org/authors/?q=ai:liu.liya"Qin, Xiaolong"https://www.zbmath.org/authors/?q=ai:qin.xiaolong"Sahu, D. R."https://www.zbmath.org/authors/?q=ai:sahu.daya-ramSummary: Based on a hybrid steepest-descent method and a splitting method, we propose a variable metric iterative algorithm, which is useful in computing the elastic net solution. A~solution theorem is established in the framework of Hilbert spaces. Numerical results are conducted on the signal processing to demonstrate the capacity and effectiveness of our algorithm.The iterative method for solving the proximal split feasibility problem with an application to LASSO problemhttps://www.zbmath.org/1483.470992022-05-16T20:40:13.078697Z"Ma, Xiaojun"https://www.zbmath.org/authors/?q=ai:ma.xiaojun"Liu, Hongwei"https://www.zbmath.org/authors/?q=ai:liu.hongwei|liu.hongwei.1"Li, Xiaoyin"https://www.zbmath.org/authors/?q=ai:li.xiaoyinSummary: In this paper, we investigate strong convergence of the iterative algorithm for solving the proximal split feasibility problem in real Hilbert spaces. The algorithm is motivated by the inertial method, the viscosity-type method and the split proximal algorithm with a self-adaptive stepsize. A~strong convergence theorem for the proposed algorithm is established without requiring firm nonexpansiveness of the involved operators. An application of our obtained results is offered. Finally, some numerical experiments are provided for illustration and comparison.On split equality minimization and fixed point problemshttps://www.zbmath.org/1483.471002022-05-16T20:40:13.078697Z"Mewomo, Oluwatosin Temitope"https://www.zbmath.org/authors/?q=ai:mewomo.oluwatosin-temitope"Ogbuisi, Ferdinard Udochukwu"https://www.zbmath.org/authors/?q=ai:ogbuisi.ferdinard-udochukwu"Okeke, Chibueze Christian"https://www.zbmath.org/authors/?q=ai:okeke.chibueze-christianSummary: In this paper, iterative algorithm for approximating a solution of a split equality minimization problem and split equality fixed point problem for demi-contractive mappings is introduced. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the solution set of a split equality minimization problem (SEMP) and the solution set of split equality fixed point problem (SEFP) for demicontractive maps. Our result does not require any compactness assumption and do not require the prior knowledge of the operator norm. Our result complements and extends some recent results in literature.Proximal point algorithm for a common of countable families of inverse strongly accretive operators and nonexpansive mappings with convergence analysishttps://www.zbmath.org/1483.471012022-05-16T20:40:13.078697Z"Promluang, Khanittha"https://www.zbmath.org/authors/?q=ai:promluang.khanittha"Sitthithakerngkiet, Kanokwan"https://www.zbmath.org/authors/?q=ai:sitthithakerngkiet.kanokwan"Kumam, Poom"https://www.zbmath.org/authors/?q=ai:kumam.poomSummary: In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well-known results from a Hilbert space to a uniformly convex and \(2\)-uniformly smooth Banach space. Finally, we establish strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.A parallel monotone hybrid algorithm for a finite family of \(G\)-nonexpansive mappings in Hilbert spaces endowed with a graph applicable in signal recoveryhttps://www.zbmath.org/1483.471022022-05-16T20:40:13.078697Z"Suantai, Suthep"https://www.zbmath.org/authors/?q=ai:suantai.suthep"Kankam, Kunrada"https://www.zbmath.org/authors/?q=ai:kankam.kunrada"Cholamjiak, Prasit"https://www.zbmath.org/authors/?q=ai:cholamjiak.prasit"Cholamjiak, Watcharaporn"https://www.zbmath.org/authors/?q=ai:cholamjiak.watcharapornSummary: In this paper, we modified the shrinking projection method with the parallel monotone hybrid method for approximating common fixed points of a finite family of \(G\)-nonexpansive mappings. We then prove a strong convergence theorem under suitable conditions in Hilbert spaces endowed with graphs. Moreover, we give some numerical examples and compare the rate of convergence of our algorithms. Finally, we provide an application to signal recovery in a situation without knowing the type of noises and demonstrate the computational performance of our algorithm in comparison to some methods. The numerical results of the comparative analysis are also discussed.Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaceshttps://www.zbmath.org/1483.471032022-05-16T20:40:13.078697Z"Timnak, Sara"https://www.zbmath.org/authors/?q=ai:timnak.sara"Naraghirad, Eskandar"https://www.zbmath.org/authors/?q=ai:naraghirad.eskandar"Hussain, Nawab"https://www.zbmath.org/authors/?q=ai:hussain.nawabSummary: In this paper, using Bregman functions, we introduce a new Halpern-type iterative algorithm for finding common zeros of finitely many maximal monotone operators and obtain a strongly convergent iterative sequence to the common zeros of these operators in a reflexive Banach space. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a finite system of equilibrium problems and null spaces of a \(\gamma\)-inverse strongly monotone mapping in a 2-uniformly convex Banach space. Some applications of our results to the solution of equations of Hammerstein-type are presented. Our scheme has an advantage that we do not use any projection of a point on the intersection of closed and convex sets which creates some difficulties in a practical calculation of the iterative sequence. So the simple construction of Halpern iteration provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective. Presented results improve and generalize many known results in the current literature.On the strong convergence theorem of the hybrid viscosity approximation method for zeros of \(m\)-accretive operators in Banach spaceshttps://www.zbmath.org/1483.471042022-05-16T20:40:13.078697Z"Truong Minh Tuyen"https://www.zbmath.org/authors/?q=ai:truong-minh-tuyen.Summary: In this paper, we give sufficient conditions of the hybrid viscosity approximation method for zeros of $m$-accretive operators in Banach spaces $E$, which was introduced by \textit{L.-C. Ceng} et al. [Numer. Funct. Anal. Optim. 33, No. 2, 142--165 (2012; Zbl 1244.47055)] (Theorem 3.2) when $E$ is a uniformly convex and uniformly smooth Banach space.A method of approximation for a zero of the sum of maximally monotone mappings In Hilbert spaceshttps://www.zbmath.org/1483.471052022-05-16T20:40:13.078697Z"Wega, Getahun Bekele"https://www.zbmath.org/authors/?q=ai:wega.getahun-bekele"Zegeye, Habtu"https://www.zbmath.org/authors/?q=ai:zegeye.habtuSummary: Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discuss its convergence. The assumption that one of the mappings is \(\alpha\)-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.The implicit midpoint rule for nonexpansive mappings in Banach spaceshttps://www.zbmath.org/1483.471062022-05-16T20:40:13.078697Z"Xu, Hong-Kun"https://www.zbmath.org/authors/?q=ai:xu.hong-kun"Alghamdi, Maryam A."https://www.zbmath.org/authors/?q=ai:alghamdi.maryam-a"Shahzad, Naseer"https://www.zbmath.org/authors/?q=ai:shahzad.naseerSummary: The implicit midpoint rule (IMR) for nonexpansive mappings is established in Banach spaces. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a uniformly convex Banach space which either satisfies Opial's property or has a Fréchet differentiable norm. Consequently, this algorithm applies in both \(\ell_p\) and \(L^p\) for \(1 < p < \infty\).A general iterative algorithm with strongly positive operators for strict pseudo-contractionshttps://www.zbmath.org/1483.471072022-05-16T20:40:13.078697Z"Xu, Wei"https://www.zbmath.org/authors/?q=ai:xu.wei.4|xu.wei.1|xu.wei.3"Wang, Yuanheng"https://www.zbmath.org/authors/?q=ai:wang.yuanhengSummary: This paper deals with a new iterative algorithm \(\{x_n\}\) with a strongly positive operator \(A\) for a \(k\)-strict pseudo-contraction \(T\) and a non-self-Lipschitzian mapping \(S\) in Hilbert spaces. Under certain appropriate conditions, the sequence \(\{x_n\}\) converges strongly to a fixed point of \(T\), which solves some variational inequality. The results here improve and extend some recent related results.Composite implicit iteration process for two finite families of generalized asymptotically quasi-nonexpansive mappingshttps://www.zbmath.org/1483.471082022-05-16T20:40:13.078697Z"Yang, Liping"https://www.zbmath.org/authors/?q=ai:yang.liping"Chen, Pan"https://www.zbmath.org/authors/?q=ai:chen.panSummary: In this article, a demiclosed principle is proved for generalized asymptotically nonexpansive mapping in a Banach space. Additionally, it proves strong convergence of a composite implicit iteration scheme to a common fixed point for two finite families of generalized asymptotically quasi-nonexpansive mappings in a nonempty closed convex subset of a uniformly convex Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. The results of this article improve and extend the corresponding results of \textit{H.-K. Xu} and \textit{R. G. Ori} [Numer. Funct. Anal. Optim. 22, No. 5--6, 767--773 (2001; Zbl 0999.47043)], \textit{Y.-Y. Zhou} and \textit{S.-S. Chang} [Numer. Funct. Anal. Optim. 23, No. 7--8, 911--921 (2002; Zbl 1041.47048)], \textit{S. S. Chang} et al. [J. Math. Anal. Appl. 313, No. 1, 273--283 (2006; Zbl 1086.47044)], \textit{L.-P. Yang} and \textit{Y. Yu} [Chin. Ann. Math., Ser. A 29, No. 6, 771--778 (2008; Zbl 1199.47306)], \textit{N. Shahzad} and \textit{H. Zegeye} [Appl. Math. Comput. 189, No. 2, 1058--1065 (2007; Zbl 1126.65054)].Approximation for the hierarchical constrained variational inequalities over the fixed points of nonexpansive semigroupshttps://www.zbmath.org/1483.471092022-05-16T20:40:13.078697Z"Zhu, Li-Jun"https://www.zbmath.org/authors/?q=ai:zhu.lijunSummary: The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point \(x^\ast\) such that \(x^\ast \in \Omega\), \(\langle (A - \gamma f)x^\ast - (I - B)Sx^\ast, x - x^\ast \rangle \geq 0\), \(\forall x \in \Omega \), where \(\Omega\) is the set of the solutions of the following variational inequality: \(x^\ast \in \mathcal{F}\), \(\langle (A - S)x^\ast, x - x^\ast \rangle \geq 0\), \(\forall x \in \mathcal{F}\), where \(A, B\) are two strongly positive bounded linear operators, \(f\) is a \(\rho\)-contraction, \(S\) is a nonexpansive mapping, and \(\mathcal{F}\) is the fixed points set of a nonexpansive semigroup \(\{T(s)\}_{s \geq 0}\). We present a double-net convergence hierarchical to some elements in \(\mathcal{F}\) which solves the above hierarchical constrained variational inequalities.Multistep-type construction of fixed point for multivalued \(\rho\)-quasi-contractive-like maps in modular function spaceshttps://www.zbmath.org/1483.471102022-05-16T20:40:13.078697Z"Akewe, Hudson"https://www.zbmath.org/authors/?q=ai:akewe.hudson"Olaoluwa, Hallowed"https://www.zbmath.org/authors/?q=ai:olaoluwa.hallowedSummary: In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map \(T\) such that \(P^T_\rho\), an associate multivalued map, is a \(\rho\)-contractive-like mapping.The implicit midpoint rule for nonexpansive mappings In 2-uniformly convex hyperbolic spaceshttps://www.zbmath.org/1483.471112022-05-16T20:40:13.078697Z"Fukhar-ud-din, H."https://www.zbmath.org/authors/?q=ai:fukharuddin.h|fukhar-ud-din.hafiz"Khan, A. R."https://www.zbmath.org/authors/?q=ai:khan.amjad-rehman|khan.abdul-rahman|khan.abdur-rauf|khan.abdul-rauf-khan|khan.ahmad-raza|khan.abdul-raim|khan.abdul-rahim|khan.abdul-rahmi|khan.asif-r|khan.atikur-rahman|khan.asif-razaSummary: The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and \(\Delta\)-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.Hybrid linesearch algorithm for pseudomonotone equilibrium problem and fixed points of Bregman quasi asymptotically nonexpansive multivalued mappingshttps://www.zbmath.org/1483.471122022-05-16T20:40:13.078697Z"Harbau, M. H."https://www.zbmath.org/authors/?q=ai:harbau.murtala-haruna"Ali, B."https://www.zbmath.org/authors/?q=ai:ali.bashirSummary: In this paper, we introduce a linesearch algorithm for solving fixed points of Bregman quasi asymptotically nonexpansive multivalued mappings and pseudomonotone equilibrium problem in reflexive Banach space. Using the linesearch method, we prove a strong convergence of the iterative scheme to a common point in the set of solutions of some equilibrium problem and common fixed point of the finite family of Bregman quasi asymptotically nonexpansive multivalued mappings with out imposing Bregman Lipschitz condition on the bifunction \(g\) as used by many authors in the extragradient method. Our results improve and generalize many recent results in the literature.Strong convergence theorem for uniformly \(L\)-Lipschitzian mapping of Gregus type in Banach spaceshttps://www.zbmath.org/1483.471132022-05-16T20:40:13.078697Z"Joshua, Olilima O."https://www.zbmath.org/authors/?q=ai:joshua.olilima-o"Adesanmi, Mogbademu A."https://www.zbmath.org/authors/?q=ai:adesanmi.mogbademu-a"Adefemi, Adeniran T."https://www.zbmath.org/authors/?q=ai:adefemi.adeniran-tSummary: In this paper, we introduce a new mapping called uniformly \(L\)-Lipschitzian mapping of Gregus type, and used the Mann iterative scheme to approximate the fixed point. A~strong convergence result for the sequence generated by the scheme is shown in real Banach space. Our result generalizes and unifies many recent results in this area of research. In addition, we give a numerical example to support our claim.Iterative methods for pseudocontractive mappings in Banach spaceshttps://www.zbmath.org/1483.471142022-05-16T20:40:13.078697Z"Jung, Jong Soo"https://www.zbmath.org/authors/?q=ai:jung.jong-sooSummary: Let \(E\) a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let \(C\) be a nonempty closed convex subset of \(E\), \(T : C \to C\) a continuous pseudocontractive mapping with \(F(T) \neq \emptyset\), and \(A : C \to C\) a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant \(k \in (0, 1)\). Let \(\{\alpha_n\}\) and \(\{\beta_n\}\) be sequences in \((0, 1)\) satisfying suitable conditions and for arbitrary initial value \(x_0 \in C\), let the sequence \(\{x_n\}\) be generated by \(x_n = \alpha_n Ax_n + \beta_n x_{n - 1} + (1 - \alpha_n - \beta_n)Tx_n\), \(n \geq 1\). If either every weakly compact convex subset of \(E\) has the fixed point property for nonexpansive mappings or \(E\) is strictly convex, then \(\{x_n\}\) converges strongly to a fixed point of \(T\), which solves a certain variational inequality related to \(A\).Approximation of fixed point of multivalued \(\rho\)-quasi-contractive mappings in modular function spaceshttps://www.zbmath.org/1483.471152022-05-16T20:40:13.078697Z"Okeke, Godwin Amechi"https://www.zbmath.org/authors/?q=ai:okeke.godwin-amechi"Khan, Safeer Hussain"https://www.zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: The purpose of this paper is to extend the recent results of \textit{G. A. Okeke} et al. [J. Funct. Spaces 2018, Article ID 1785702, 9 p. (2018; Zbl 1442.47062)] to the class of multivalued \(\rho\)-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of \(T\)-stability, almost \(T\)-stability and summably almost \(T\)-stability to modular function spaces and give some results.Strong convergence theorems for semigroups of asymptotically nonexpansive mappings in Banach spaceshttps://www.zbmath.org/1483.471162022-05-16T20:40:13.078697Z"Sahu, D. R."https://www.zbmath.org/authors/?q=ai:sahu.daya-ram"Wong, Ngai-Ching"https://www.zbmath.org/authors/?q=ai:wong.ngai-ching"Yao, Jen-Chih"https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: Let \(X\) be a real reflexive Banach space with a weakly continuous duality mapping \(J_\varphi\). Let \(C\) be a nonempty weakly closed star-shaped (with respect to \(u\)) subset of \(X\). Let \(\mathcal{F} = \{T(t) : t \in [0, +\infty]\}\) be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of \(C\), which is uniformly continuous at zero. We will show that the implicit iteration scheme \(y_n = \alpha_n u + (1 - \alpha_n)T(t_n)y_n\), for all \(n \in \mathbb N\), converges strongly to a common fixed point of the semigroup \(\mathcal F\) for some suitably chosen parameters \(\{\alpha_n\}\) and \(\{t_n\}\). Our results extend and improve corresponding ones of \textit{T. Suzuki} [Proc. Am. Math. Soc. 131, No. 7, 2133--2136 (2003; Zbl 1031.47038)], \textit{H.-K. Xu} [Bull. Aust. Math. Soc. 72, No. 3, 371--379 (2005; Zbl 1095.47016)], and \textit{H. Zegeye} and \textit{N. Shahzad} [Numer. Funct. Anal. Optim. 30, No. 7--8, 833--848 (2009; Zbl 1177.47084)].Weak convergence theorems for two asymptotically quasi-nonexpansive non-self mappings in uniformly convex Banach spaceshttps://www.zbmath.org/1483.471172022-05-16T20:40:13.078697Z"Saluja, G. S."https://www.zbmath.org/authors/?q=ai:saluja.gurucharan-singhSummary: The purpose of this paper is to establish some weak convergence theorems of modified two-step iteration process with errors for two asymptotically quasi-nonexpansive non-self mappings in the setting of real uniformly convex Banach spaces if \(E\) satisfies Opial's condition or the dual \(E^{\ast}\) of \(E\) has the Kadec-Klee property. Our results extend and improve some known corresponding results from the existing literature.Implicit iterative scheme for a countable family of nonexpansive mappings in 2-uniformly smooth Banach spaceshttps://www.zbmath.org/1483.471182022-05-16T20:40:13.078697Z"Tian, Ming"https://www.zbmath.org/authors/?q=ai:tian.ming"Jin, Xin"https://www.zbmath.org/authors/?q=ai:jin.xin.1Summary: Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator \(F : x_n = \alpha_n \gamma V(x_n) + \beta_n x_{n - 1} + ((1 - \beta_n)I - \alpha_n \mu F)T_n x_n\) and get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaceshttps://www.zbmath.org/1483.471192022-05-16T20:40:13.078697Z"Ullah, Kifayat"https://www.zbmath.org/authors/?q=ai:ullah.kifayat"Khan, Muhammad Safi Ullah"https://www.zbmath.org/authors/?q=ai:khan.muhammad-safi-ullah"Muhammad, Naseer"https://www.zbmath.org/authors/?q=ai:muhammad.naseer"Ahmad, Junaid"https://www.zbmath.org/authors/?q=ai:ahmad.junaidThe theta method for nonexpansive mappingshttps://www.zbmath.org/1483.471202022-05-16T20:40:13.078697Z"Xu, Hong-Kun"https://www.zbmath.org/authors/?q=ai:xu.hong-kun"Alghamdi, Maryam A."https://www.zbmath.org/authors/?q=ai:alghamdi.maryam-a"Shahzad, Naseer"https://www.zbmath.org/authors/?q=ai:shahzad.naseerSummary: The $\theta$-method for a general nonexpansive mapping $T$ in a Hilbert space is introduced. This method generates a sequence in an implicit manner. It is proved that this sequence converges weakly to a fixed point of $T$ under a divergence condition on the parameters that define the method. Applications to Fredholm integral equations, nonlinear evolution equations and constrained optimization problems are included.Strong convergence theorems for a common fixed point of a family of asymptotically \(k\)-strict pseudocontractive mappingshttps://www.zbmath.org/1483.471212022-05-16T20:40:13.078697Z"Zegeye, H."https://www.zbmath.org/authors/?q=ai:zegeye.habtu"Shahzad, N."https://www.zbmath.org/authors/?q=ai:shahzad.naseer|shahzad.nasserSummary: We provide an iterative process which converges strongly to a common fixed point of finite family of asymptotically \(k\)-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.Parabolic variational inequalities with generalized reflecting directionshttps://www.zbmath.org/1483.471222022-05-16T20:40:13.078697Z"Rotenstein, Eduard"https://www.zbmath.org/authors/?q=ai:rotenstein.eduard-paulSummary: We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type:
\[
y'(t)+ Ay(t)+\Theta(t,y(t))\partial\varphi(y(t))\ni f(t,y(t)),\,y(0) = y_{0},\quad t\in[0,T],
\]
where \(A\) is a linear self-adjoint operator, \(\partial\varphi\) is the subdifferential operator of a proper lower semicontinuous convex function \(\varphi\) defined on a suitable Hilbert space, and \(\Theta\) is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.Almost contractive maps between \(C^{\ast}\)-algebras with applications to Fourier algebrashttps://www.zbmath.org/1483.471232022-05-16T20:40:13.078697Z"Ricard, É."https://www.zbmath.org/authors/?q=ai:ricard.eric"Roydor, J."https://www.zbmath.org/authors/?q=ai:roydor.jeanSummary: We prove that unital almost contractive maps between \(C^\ast\)-algebras enjoy approximately certain properties of unital positive maps (such as selfadjointness or the Kadison-Schwarz inequality). Our main application is a description of almost contractive homomorphisms between Fourier algebras and Fourier-Stieltjes algebras: they are actually contractive if their norm is smaller than 1.00018. For surjective isomorphisms of Fourier algebras, the bound 1.0005 is sufficient in order to obtain an isometry.Unbounded graph-Laplacians in energy space, and their extensionshttps://www.zbmath.org/1483.471242022-05-16T20:40:13.078697Z"Jorgensen, Palle E. T."https://www.zbmath.org/authors/?q=ai:jorgensen.palle-e-tSummary: Our purpose is to develop computational tools for determining spectra for operators associated with infinite weighted graphs. While there is a substantial literature concerning graph-Laplacians on infinite networks, much less developed is the distinction between the operator theory for the \(\ell^2\) space of the set \(V\) of vertices vs the case when the Hilbert space is defined by an energy form. A~network is a triple \((V,E,c)\) where \(V\) is a (typically countable infinite) set of vertices in a graph, with \(E\) denoting the set of edges. The function \(c\) is defined on \(E\). It is given at the outset, symmetric and positive on \(E\). We introduce a graph-Laplacian \(\Delta\), and an energy Hilbert space \(\mathcal H_E\) (both depending on \(c\)). While it is known that \(\Delta\) is essentially selfadjoint on its natural domain in \(\ell^2(V)\), its realization in \(\mathcal H_E\) is not. We give a characterization of the Friedrichs extension of the \(\mathcal H_E\)-Laplacian, and prove a formula for its computation. We obtain several corollaries regarding the diagonalization of infinite matrices. To every weighted finite-interaction countable infinite graph there is a naturally associated infinite banded matrix. With the use of the Friedrichs spectral resolution, we obtain a diagonalization formula for this family of infinite matrices. With examples we give concrete illustrations of both spectral types, and spectral multiplicities.Differential equation approximations of stochastic network processes: an operator semigroup approachhttps://www.zbmath.org/1483.471252022-05-16T20:40:13.078697Z"Bátkai, András"https://www.zbmath.org/authors/?q=ai:batkai.andras"Kiss, Istvan Z."https://www.zbmath.org/authors/?q=ai:kiss.istvan-z"Sikolya, Eszter"https://www.zbmath.org/authors/?q=ai:sikolya.eszter"Simon, Péter L."https://www.zbmath.org/authors/?q=ai:simon.peter-lSummary: The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view, the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (\(N\)). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as \(N\) tends to infinity. Using only elementary semigroup theory, we can prove the order \(\mathcal{O}(1/N)\) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a~new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.From backward approximations to Lagrange polynomials in discrete advection-reaction operatorshttps://www.zbmath.org/1483.471262022-05-16T20:40:13.078697Z"Solis, Francisco J."https://www.zbmath.org/authors/?q=ai:solis.francisco-javier"Barradas, Ignacio"https://www.zbmath.org/authors/?q=ai:barradas.ignacio"Juarez, Daniel"https://www.zbmath.org/authors/?q=ai:juarez.danielSummary: In this work we introduce a family of operators called discrete advection-reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection-reaction-diffusion partial differential equations. They consist of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.On Heisenberg's inequality and Bell's inequalityhttps://www.zbmath.org/1483.471272022-05-16T20:40:13.078697Z"Nagasawa, Masao"https://www.zbmath.org/authors/?q=ai:nagasawa.masaoFrom the introduction: There is a well-known proof of Heisenberg-Kennard inequality by \textit{H. P. Robertson} [Phys. Rev. 34, No. 1, 163--164 (1929; \url{doi:10.1103/PhysRev.34.163})] who used Schwarz's inequality.
We prove, however, that Heisenberg-Kennard inequality follows from Robertson's inequality only in a very special case, but does not follow in general.
Therefore, Heisenberg's uncertainty principle in quantum mechanics is false.
Bell's claim ``no local hidden variable model can explain the quantum mechanical correlation'' in \textit{J. S. Bell} [Phys. Phys. Fiz. 1, No. 3, 195--202 (1964; \url{doi:10.1103/PhysicsPhysiqueFizika.1.195})] is also shown to be false. Bell's claim was based on Bell's inequality. We will show that Bell's inequality concerns neither locality nor non-locality at all. Hence Bell's claim doesn't follow from Bell's inequality. We show in addition that Bell's inequality holds only under Bell's additional dependence condition, but does not follow in general. We will moreover give a local spin correlation model, which is a counter-example against Bell's claim.\(S\)-iterative algorithm for solving variational inequalitieshttps://www.zbmath.org/1483.490142022-05-16T20:40:13.078697Z"Ertürk, Müzeyyen"https://www.zbmath.org/authors/?q=ai:erturk.muzeyyen"Gürsoy, Faik"https://www.zbmath.org/authors/?q=ai:gursoy.faik"Şimşek, Necip"https://www.zbmath.org/authors/?q=ai:simsek.necipSummary: In this paper we propose an \(S\)-iterative algorithm for finding the common element of the set of solutions of the variational inequalities and the set of fixed points of nonexpansive mappings. We study the convergence criteria of the mentioned algorithm under some mild conditions. As an application, a modified algorithm is suggested to solve convex minimization problems. Numerical examples are given to validate the theoretical findings obtained herein. Our results may be considered as an improvement, refinement and complement of the previously known results.Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systemshttps://www.zbmath.org/1483.490292022-05-16T20:40:13.078697Z"Sumin, Vladimir Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.v-i"Sumin, Mikhail Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.mikhail-iosifovichSummary: We consider the regularization of the \textit{classical optimality conditions} (COCs) -- the Lagrange principle and the Pontryagin maximum principle -- in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space \(L^m_2\), the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They ``overcome'' the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.Monotone systems involving variable-order nonlocal operatorshttps://www.zbmath.org/1483.490402022-05-16T20:40:13.078697Z"Yangari, Miguel"https://www.zbmath.org/authors/?q=ai:yangari.miguelIn this paper, the author investigates the existence (via Perron's method) and uniqueness of bounded viscosity solutions for some monotone systems of parabolic Hamilton-Jacobi type. Here, the diffusion term is determined by variable-order nonlocal operators whose kernels depend on the space-time variable.
Reviewer: Savin Treanta (Bucureşti)Existence of coincidence and common fixed points for a sequence of mappings in quasi partial metric spaceshttps://www.zbmath.org/1483.540292022-05-16T20:40:13.078697Z"Dhawan, Pooja"https://www.zbmath.org/authors/?q=ai:dhawan.pooja"Gupta, Vishal"https://www.zbmath.org/authors/?q=ai:gupta.vishal"Kaur, Jatinderdeep"https://www.zbmath.org/authors/?q=ai:kaur.jatinderdeepSummary: Till now, there exists enormous literature showing existence of fixed points using expansive mappings. But the existence of common and coincidence fixed points for a sequence of functions using expansive mappings is still uncharted. In the present article, some coincidence and common fixed point results for a sequence of mappings satisfying generalized weakly expansive conditions in the setting of quasi partial metric spaces have been investigated. The effectiveness of obtained results have been verified with the help of some comparative examples.Common fixed point results for class of set-contraction mappings endowed with a directed graphhttps://www.zbmath.org/1483.540302022-05-16T20:40:13.078697Z"Latif, Abdul"https://www.zbmath.org/authors/?q=ai:latif.abdul"Nazir, Talat"https://www.zbmath.org/authors/?q=ai:nazir.talat"Kutbi, Marwan Amin"https://www.zbmath.org/authors/?q=ai:kutbi.marwan-aminSummary: We present common fixed point results of finite family of set-valued mappings satisfying generalized graphic contractions on set-valued domain endowed with a directed graph. Some examples are presented to support the results proved therein. Consequently, our results unify, improve and generalize various comparable known results.On the KKM theory on ordered spaceshttps://www.zbmath.org/1483.540322022-05-16T20:40:13.078697Z"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since \textit{C. D. Horvath} and \textit{J. V. Llinares Cisar} [J. Math. Econ. 25, No. 3, 291--306 (1996; Zbl 0852.90006)] began to study maximal elements and fixed points for binary relations on topological ordered spaces, there have appeared many works related to the KKM theory on such spaces by several authors. Independently to these works, we began to study the KKM theory on abstract convex spaces [the author, Nonlinear Anal. Forum 11, No. 1, 67--77 (2006; Zbl 1120.47038)]. Our aim in the present paper is to extend the known results on topological ordered spaces to the corresponding ones on abstract convex spaces.JS-Prešić contractive mappings in extended modular \(S\)-metric spaces and extended fuzzy \(S\)-metric spaces with an applicationhttps://www.zbmath.org/1483.540342022-05-16T20:40:13.078697Z"Rezaee, M. M."https://www.zbmath.org/authors/?q=ai:rezaee.mohammad-mahdi"Sedghi, S."https://www.zbmath.org/authors/?q=ai:sedghi.shaban"Parvaneh, V."https://www.zbmath.org/authors/?q=ai:parvaneh.vahidSummary: In this paper, we introduce the concept of extended modular \(S\)-metric spaces which induce the notion of extended fuzzy \(S\)-metric spaces and is a generalization of some classes of metric type spaces. We obtain some results for JS-Prešić contractive mappings in this new setting and in the related fuzzy setting. In fact, we obtain the Prešić fixed points via an easier way than the previous methods. An application in integral equations will support our results.Common fixed point theorems for T-Hardy-Rodgers contraction mappings in complete cone b-metric spaces with an applicationhttps://www.zbmath.org/1483.540382022-05-16T20:40:13.078697Z"Shashi, Pauline"https://www.zbmath.org/authors/?q=ai:shashi.pauline"Kumar, Santosh"https://www.zbmath.org/authors/?q=ai:kumar.santosh|kumar.santosh.3|kumar.santosh.2|kumar.santosh.1|kumar.santosh.4Summary: This paper presents some fixed point theorems for T-Hardy-Rodgers contraction mappings in complete cone b-metric spaces without the assumption of normality conditions. We prove the existence of the common fixed point in complete cone b-metric spaces for the continuous self mappings. Our results generalize many recent known results in metric spaces, b-metric spaces and cone metric spaces in the literature.Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical componentshttps://www.zbmath.org/1483.580052022-05-16T20:40:13.078697Z"Baldare, Alexandre"https://www.zbmath.org/authors/?q=ai:baldare.alexandre"Côme, Rémi"https://www.zbmath.org/authors/?q=ai:come.remi"Lesch, Matthias"https://www.zbmath.org/authors/?q=ai:lesch.matthias"Nistor, Victor"https://www.zbmath.org/authors/?q=ai:nistor.victorSummary: Let \(\Gamma\) be a compact group acting on a smooth, compact manifold \(M\), let \(P\in\psi^m(M;E_0,E_1)\) be a \(\Gamma\)-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles \(E_i\to M\), \(i=0,1\), and let \(\alpha\) be an irreducible representation of the group \(\Gamma\). Then \(P\) induces a map \(\pi_\alpha(P):H^s(M;E_0)_\alpha\to H^{s-m}(M;E_1)_\alpha\) between the \(\alpha\)-isotypical components of the corresponding Sobolev spaces of sections. When \(\Gamma\) is finite, we explicitly characterize the operators \(P\) for which the map \(\pi_\alpha(P)\) is Fredholm in terms of the principal symbol of \(P\) and the action of \(\Gamma\) on the vector bundles \(E_i\). When \(\Gamma=\{1\}\), that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol \(C^*\)-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.Persistence exponents in Markov chainshttps://www.zbmath.org/1483.600412022-05-16T20:40:13.078697Z"Aurzada, Frank"https://www.zbmath.org/authors/?q=ai:aurzada.frank"Mukherjee, Sumit"https://www.zbmath.org/authors/?q=ai:mukherjee.sumit"Zeitouni, Ofer"https://www.zbmath.org/authors/?q=ai:zeitouni.oferSummary: We prove the existence of the \textit{persistence exponent}
\[
\log\lambda :=\lim\limits_{n\to \infty}\frac{1}{n}\log{\mathbb{P}_{\mu}}({X_0}\in S,\dots,{X_n}\in S)
\]
for a class of time homogeneous Markov chains \(\{X_i\}_{i\ge 0}\) taking values in a Polish space, where \(S\) is a Borel measurable set and \(\mu\) is an initial distribution. Focusing on the case of \(\mathrm{AR}(p)\) and \(\mathrm{MA}(q)\) processes with \(p,q\in\mathbb{N}\) and continuous innovation distribution, we study the existence of \(\lambda\) and its continuity in the parameters of the AR and MA processes, respectively, for \(S=\mathbb{R}_{\ge 0}\). For AR processes with log-concave innovation distribution, we prove the strict monotonicity of \(\lambda\). Finally, we compute new explicit exponents in several concrete examples.Convolution algebra for extended Feller convolutionhttps://www.zbmath.org/1483.600712022-05-16T20:40:13.078697Z"Lee, Wha-Suck"https://www.zbmath.org/authors/?q=ai:lee.wha-suck"Le Roux, Christiaan"https://www.zbmath.org/authors/?q=ai:le-roux.christiaanThe paper considers the continuous analogue to the problem of intertwined Markov processes with extended Chapman-Kolmogorov's equation in the form of two distinct continuous state spaces and two homogeneous Markov processes, which models random transitions within a continuum of ``life'' states and from the ``life'' states to a continuum of ``death'' states. The absorbing barrier is modelled as a continuum as ``death'' states for the case of a single ``death'' state.
To handle two-dimensional uni-directional homogeneous stochastic kernels arising in this problem, the authors use recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $\mathbb{C}^2$-valued admissible homomorphisms. For an adequate operator representation of such kernels, the algebra product, which is a non-commutative extension of the Feller convolution, is needed.
Reviewer: Anatoliy Swishchuk (Calgary)Random quasi-periodic paths and quasi-periodic measures of stochastic differential equationshttps://www.zbmath.org/1483.600792022-05-16T20:40:13.078697Z"Feng, Chunrong"https://www.zbmath.org/authors/?q=ai:feng.chunrong"Qu, Baoyou"https://www.zbmath.org/authors/?q=ai:qu.baoyou"Zhao, Huaizhong"https://www.zbmath.org/authors/?q=ai:zhao.huaizhongSummary: In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.Moments of the 2D SHE at criticalityhttps://www.zbmath.org/1483.600932022-05-16T20:40:13.078697Z"Gu, Yu"https://www.zbmath.org/authors/?q=ai:gu.yu.1"Quastel, Jeremy"https://www.zbmath.org/authors/?q=ai:quastel.jeremy"Tsai, Li-Cheng"https://www.zbmath.org/authors/?q=ai:tsai.li-chengSummary: We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius \(\varepsilon\to 0 \), we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit nontrivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of \textit{J. Dimock} and \textit{S. G. Rajeev} [J. Phys. A, Math. Gen. 37, No. 39, 9157--9173 (2004; Zbl 1067.81024)] to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.Variational inequality over the set of common solutions of a system of bilevel variational inequality problem with applicationshttps://www.zbmath.org/1483.650822022-05-16T20:40:13.078697Z"Eslamian, Mohammad"https://www.zbmath.org/authors/?q=ai:eslamian.mohammadSummary: In this paper, we study variational inequality problem over the set of common solutions of a system of bilevel variational inequality problem. We present a new and efficient iterative method for solving this problem and establish its strong convergence. As applications, we use our algorithm for solving the multiple set split variational inequality problem, the hierarchical variational inequality problem, the bilevel variational inequality problem and hierarchical minimization problem.Hybrid iteration method for fixed points of asymptotically \(\phi\)-demicontractive maps in real Hilbert spaceshttps://www.zbmath.org/1483.650842022-05-16T20:40:13.078697Z"Jim, Uko Sunday"https://www.zbmath.org/authors/?q=ai:jim.uko-sundaySummary: A strong convergence theorem of Hybrid iteration method to fixed points of asymptotically \(\phi\)-demicontractive mapping is proved in real Hilbert spaces. Our results extend, generalize and complement the results of \textit{L. Wang} [Fixed Point Theory Appl. 2007, Article ID 28619, 8 p. (2007; Zbl 1159.47052)], \textit{M. O. Osilike} et al. [Fixed Point Theory Appl. 2007, Article ID 64306, 7 p. (2007; Zbl 1156.47311)], and extend several others from asymptotically demicontractive to the more general class of asymptotically \(\phi\)-demicontractive maps (see for example [\textit{D. I. Igbokwe}, JIPAM, J. Inequal. Pure Appl. Math. 3, No. 1, Paper No. 3, 11p. (2002; Zbl 1009.47061); \textit{M. O. Osilike}, Indian J. Pure Appl. Math. 29, No. 12, 1291--1300 (1998; Zbl 0927.47039); \textit{M. O. Osilike} et al., Panam. Math. J. 12, No. 2, 77--88 (2002; Zbl 1018.47047)]).Penalty-based smoothness conditions in convex variational regularizationhttps://www.zbmath.org/1483.650872022-05-16T20:40:13.078697Z"Hofmann, Bernd"https://www.zbmath.org/authors/?q=ai:hofmann.bernd"Kindermann, Stefan"https://www.zbmath.org/authors/?q=ai:kindermann.stefan"Mathé, Peter"https://www.zbmath.org/authors/?q=ai:mathe.peterSummary: The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.An inertial Tseng's extragradient method for solving multi-valued variational inequalities with one projectionhttps://www.zbmath.org/1483.651062022-05-16T20:40:13.078697Z"Fan, Changjie"https://www.zbmath.org/authors/?q=ai:fan.changjie"Zhang, Ruirui"https://www.zbmath.org/authors/?q=ai:zhang.ruirui"Chen, Shenglan"https://www.zbmath.org/authors/?q=ai:chen.shenglanSummary: We introduce an inertial Tseng's extragradient method for solving multi-valued variational inequalities, in which only one projection is needed at each iteration. We also obtain the strong convergence results of the proposed algorithm, provided that the multi-valued mapping is continuous and pseudomonotone with nonempty compact convex values. Moreover, numerical simulation results illustrate the efficiency of our method when compared to existing methods.Proximal extrapolated gradient methods for variational inequalitieshttps://www.zbmath.org/1483.651072022-05-16T20:40:13.078697Z"Malitsky, Yu"https://www.zbmath.org/authors/?q=ai:malitsky.yu-v|malitsky.yuriSummary: The paper concerns with novel first-order methods for monotone variational inequalities. They use a very simple linesearch procedure that takes into account a local information of the operator. Also, the methods do not require Lipschitz continuity of the operator and the linesearch procedure uses only values of the operator. Moreover, when the operator is affine our linesearch becomes very simple, namely, it needs only simple vector-vector operations. For all our methods, we establish the ergodic convergence rate. In addition, we modify one of the proposed methods for the case of a composite minimization. Preliminary results from numerical experiments are quite promising.An inexact Newton method with inner preconditioned CG for non-uniformly monotone elliptic problemshttps://www.zbmath.org/1483.651762022-05-16T20:40:13.078697Z"Borsos, Benjámin"https://www.zbmath.org/authors/?q=ai:borsos.benjaminSummary: The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.Numerical solutions of non-linear system of higher order Volterra integro-differential equations using generalized STWS techniquehttps://www.zbmath.org/1483.652312022-05-16T20:40:13.078697Z"Sekar, R. Chandra Guru"https://www.zbmath.org/authors/?q=ai:sekar.r-chandra-guru"Murugesan, K."https://www.zbmath.org/authors/?q=ai:murugesan.kSummary: In this article, we deal with non-linear system of higher order Volterra integro-differential equations and their numerical solutions using the Single Term Walsh Series (STWS) method. The connections between STWS coefficients of the unknown functions and its derivatives are derived. The non-linear system of Volterra integro-differential equations are converted into a system of non-linear algebraic equations using the Single Term Walsh Series coefficients. Solving these system of algebraic equations, we obtain the discrete numerical solutions of the non-linear Volterra integro-differential equations. Numerical examples are presented to show the efficiency and applicability of this STWS method for solving the non-linear system of higher order Volterra integro-differential equations.Affine quantum harmonic analysishttps://www.zbmath.org/1483.810052022-05-16T20:40:13.078697Z"Berge, Eirik"https://www.zbmath.org/authors/?q=ai:berge.eirik"Berge, Stine Marie"https://www.zbmath.org/authors/?q=ai:berge.stine-marie"Luef, Franz"https://www.zbmath.org/authors/?q=ai:luef.franz"Skrettingland, Eirik"https://www.zbmath.org/authors/?q=ai:skrettingland.eirikSummary: We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency representations. In the process, we develop a notion of admissibility for operators and extend well known results to the operator setting. A major theme of the paper is the interaction between operator convolutions, affine Weyl quantization, and admissibility.Thermal entanglement in \(2 \times 3\) Heisenberg chains via distance between stateshttps://www.zbmath.org/1483.810312022-05-16T20:40:13.078697Z"Silva, Saulo L. L."https://www.zbmath.org/authors/?q=ai:silva.saulo-l-lSummary: Most of the work involving entanglement measurement focuses on systems that can be modeled by two interacting qubits. This is due to the fact that there are few studies presenting entanglement analytical calculations in systems with spins \(s > 1/2\). In this paper, we present for the first time an analytical way of calculating thermal entanglement in a dimension \(2 \otimes 3\) Heisenberg chain through the distance between states. We use the Hilbert-Schmidt norm to obtain entanglement. The result obtained can be used to calculate entanglement in chains with spin-1/2 coupling with spin-1, such as ferrimagnetic compounds as well as compounds with dimer-trimer coupling.Two-dimensional Dirac operators with interactions on unbounded smooth curveshttps://www.zbmath.org/1483.810642022-05-16T20:40:13.078697Z"Rabinovich, V."https://www.zbmath.org/authors/?q=ai:rabinovich.vladimir-s|rabinovich.vladimir-lSummary: We consider the 2D Dirac operator with singular potentials
\[ \mathfrak{D}_{\boldsymbol{A},\Phi,Q_{\sin}}\boldsymbol{u}(x)=\left( \mathfrak{D}_{\boldsymbol{A},\Phi}+Q_{\sin}\right) \boldsymbol{u} (x),\quad x\in\mathbb{R}^2,\tag{1}\]
where
\[\mathfrak{D}_{\boldsymbol{a},\Phi}= {\displaystyle\sum\limits_{j=1}^2} \sigma_j\left( i\partial_{x_j}+a_j\right) +\sigma_3m+\Phi I_2;\tag{2}\]
here \(\sigma_j\), \(j=1,2,3,\) are Pauli matrices, \( \boldsymbol{a=}(a_1,a_2)\) is the magnetic potential with \(a_j\in L^{\infty}(\mathbb{R}^2)\), \(\Phi\in L^{\infty}(\mathbb{R)}\) is the electrostatic potential, \(Q_{\sin} =Q\delta_{\Gamma}\) is the singular potential with the strength matrix \(Q=\left( Q_{ij}\right)_{i,j=1}^2\), and \(\delta_{\Gamma}\) is the delta-function with support on a \(C^2\)-curve \(\Gamma \), which is the common boundary of the domains \(\Omega_{\pm}\subset\mathbb{R}^2\). We associate with the formal Dirac operator \(\mathfrak{D}_{\boldsymbol{a},\Phi,Q_{\sin}}\) an unbounded operator \(\mathscr{D}_{\boldsymbol{A,}\Phi,Q}\) in \(L^2 (\mathbb{R}^2,\mathbb{C}^2)\) generated by \(\mathfrak{D}_{\boldsymbol{a} ,\Phi}\) with a domain in \(H^1(\Omega_+,\mathbb{C}^2)\oplus H^1 (\Omega_-,\mathbb{C}^2)\) consisting of functions satisfying interaction conditions on \(\Gamma.\) We study the self-adjointness of the operator \(\mathscr{D}_{\boldsymbol{A,}\Phi,Q}\) and its essential spectrum for potentials and curves \(\Gamma\) slowly oscillating at infinity. We also study the splitting of the interaction problems into two boundary problems describing the confinement of particles in the domains \(\Omega_{\pm}\).Spectral and scattering theory of one-dimensional coupled photonic crystalshttps://www.zbmath.org/1483.810662022-05-16T20:40:13.078697Z"De Nittis, G."https://www.zbmath.org/authors/?q=ai:de-nittis.giuseppe"Moscolari, M."https://www.zbmath.org/authors/?q=ai:moscolari.massimo"Richard, S."https://www.zbmath.org/authors/?q=ai:richard.serge"Tiedra de Aldecoa, R."https://www.zbmath.org/authors/?q=ai:tiedra-de-aldecoa.rafaelThe IDS and asymptotic of the largest eigenvalue of random Schrödinger operators with decaying random potentialhttps://www.zbmath.org/1483.810672022-05-16T20:40:13.078697Z"Dolai, Dhriti Ranjan"https://www.zbmath.org/authors/?q=ai:dolai.dhriti-ranjanQuantum graphs on radially symmetric antitreeshttps://www.zbmath.org/1483.810742022-05-16T20:40:13.078697Z"Kostenko, Aleksey"https://www.zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://www.zbmath.org/authors/?q=ai:nicolussi.noemaIn the present study the authors mainly focused their attention on antitrees from the perspective of quantum graphs and discussed a detailed spectral analysis of the Kirchhoff Laplacian on radially symmetric antitrees. Antitrees come into sight in the investigation of discrete Laplacians and attracted a noteworthy attention especially after the work of \textit{K.-T. Sturm} [J. Reine Angew. Math. 456, 173--196 (1994; Zbl 0806.53041)]. Also, Kostenko and Nicolussi considered the approach intorudced by [\textit{V. A. Mikhailets}, Funct. Anal. Appl. 30, No. 2, 144--146 (1996; Zbl 0874.34069); translation from Funkts. Anal. Prilozh. 30, No. 2, 90--93 (1996); \textit{B. Muckenhoupt}, Stud. Math. 44, 31--38 (1972; Zbl 0236.26015)] for radially symmetric trees and used some ideas from [\textit{J. Breuer} and \textit{N. Levi}, Ann. Henri Poincaré 21, No. 2, 499--537 (2020; Zbl 1432.05061)], where discrete Laplacians on radially symmetric ``weighted'' graphs have been analyzed. To summarize in general terms, in this paper, after recalling some necessary definitions and presenting an hypothesis, the authors studied characterization of self-adjointness and a complete description of self-adjoint extensions, spectral gap estimates and spectral types (discrete, singular and absolutely continuous spectrum). Next, they demonstrated their main results by considering two special classes of antitrees: (i) antitrees with exponentially increasing sphere numbers and (ii) antitrees with polynomially increasing sphere numbers.
Reviewer: Mustafa Salti (Mersin)On the spectrum discreteness for the magnetic Schrödinger operator on quantum graphshttps://www.zbmath.org/1483.810762022-05-16T20:40:13.078697Z"Popov, Igor Y."https://www.zbmath.org/authors/?q=ai:popov.igor-yu"Belolipetskaia, Anna G."https://www.zbmath.org/authors/?q=ai:belolipetskaia.anna-gSummary: The aim of this work is to study the discreteness of the spectrum of the Schrödinger operator on infinite quantum graphs in a magnetic field. The problem was solved on a set of quantum graphs of a special kind.The divergence of Van Hove's model and its consequenceshttps://www.zbmath.org/1483.810952022-05-16T20:40:13.078697Z"Sbisà, Fulvio"https://www.zbmath.org/authors/?q=ai:sbisa.fulvioSummary: We study a regularized version of Van Hove's 1952 model, in which a quantum field interacts linearly with sources of finite width lying at fixed positions. We show that the central result of Van Hove's 1952 paper on the foundations of Quantum Field Theory, the orthogonality between the spaces of state vectors which correspond to different values of the parameters of the theory, disappears when a well-defined model is considered. We comment on the implications of our results for the contemporary relevance of Van Hove's article.Generalized eigenfunctions for quantum walks via path counting approachhttps://www.zbmath.org/1483.811342022-05-16T20:40:13.078697Z"Komatsu, Takashi"https://www.zbmath.org/authors/?q=ai:komatsu.takashi"Konno, Norio"https://www.zbmath.org/authors/?q=ai:konno.norio"Morioka, Hisashi"https://www.zbmath.org/authors/?q=ai:morioka.hisashi"Segawa, Etsuo"https://www.zbmath.org/authors/?q=ai:segawa.etsuoAbsence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fieldshttps://www.zbmath.org/1483.811522022-05-16T20:40:13.078697Z"Dimassi, Mouez"https://www.zbmath.org/authors/?q=ai:dimassi.mouez"Kawamoto, Masaki"https://www.zbmath.org/authors/?q=ai:kawamoto.masaki"Petkov, Vesselin"https://www.zbmath.org/authors/?q=ai:petkov.vesselin-mOptimization over trace polynomialshttps://www.zbmath.org/1483.901062022-05-16T20:40:13.078697Z"Klep, Igor"https://www.zbmath.org/authors/?q=ai:klep.igor"Magron, Victor"https://www.zbmath.org/authors/?q=ai:magron.victor"Volčič, Jurij"https://www.zbmath.org/authors/?q=ai:volcic.jurijSummary: Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme [\textit{M. Navascués} et al., ``A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations'', New J. Phys. 10, No. 7, 073013 (2008; \url{doi:10.1088/1367-2630/10/7/073013})] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.Copositivity for 3rd-order symmetric tensors and applicationshttps://www.zbmath.org/1483.901072022-05-16T20:40:13.078697Z"Liu, Jiarui"https://www.zbmath.org/authors/?q=ai:liu.jiarui"Song, Yisheng"https://www.zbmath.org/authors/?q=ai:song.yisheng|song.yisheng.1Summary: The strict copositivity of 4th-order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) copositivity of 4th-order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided several analytically sufficient conditions for the copositivity of 3rd-order 2-dimensional (3-dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th-order tensors, the analytically sufficient conditions of copositivity are proved for 4th-order 2-dimensional and 3-dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for \(\mathbb{Z}_3\) scalar dark matter.Null controllability of a nonlinear age structured model for a two-sex populationhttps://www.zbmath.org/1483.921142022-05-16T20:40:13.078697Z"Traoré, Amidou"https://www.zbmath.org/authors/?q=ai:traore.amidou"Sougué, Okana S."https://www.zbmath.org/authors/?q=ai:sougue.okana-s"Simporé, Yacouba"https://www.zbmath.org/authors/?q=ai:simpore.yacouba"Traoré, Oumar"https://www.zbmath.org/authors/?q=ai:traore.oumarSummary: This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if \(A\) is the maximal age, a time interval of duration \(A\) after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani's fixed-point theorem.Controllable systems with vanishing energyhttps://www.zbmath.org/1483.930392022-05-16T20:40:13.078697Z"Zabczyk, Jerzy"https://www.zbmath.org/authors/?q=ai:zabczyk.jerzySummary: The paper surveys results on controllable systems with vanishing energy introduced by
\textit{E. Priola} and \textit{J. Zabczyk} [SIAM J. Control Optim. 42, No. 3, 1013--1032 (2003; Zbl 1050.93010)].
Applications to space travels and to partial differential equations are discussed.