Recent zbMATH articles in MSC 47Bhttps://www.zbmath.org/atom/cc/47B2022-05-16T20:40:13.078697ZWerkzeugSharp 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)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 \(D^\alpha\).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.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)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.krzysztofKoopman 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.costasBi-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)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)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)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 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)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.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)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)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)].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.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.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.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.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.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.