Recent zbMATH articles in MSC 47https://www.zbmath.org/atom/cc/472021-04-16T16:22:00+00:00WerkzeugOn singular perturbations of quantum dynamical semigroups.https://www.zbmath.org/1456.470142021-04-16T16:22:00+00:00"Holevo, A. S."https://www.zbmath.org/authors/?q=ai:holevo.alexander-sSummary: We consider two examples of quantum dynamical semigroups obtained by singular perturbations of a standard generator which are special case of unbounded completely positive perturbations studied in detail in [\textit{A. S. Holevo}, Izv. Math. 59, No. 6, 1311--1325 (1995; Zbl 0877.47026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 6, 207--222 (1995)]. In Section~2, we propose a generalization of an example in [\textit{A. S. Holevo}, Izv. Math. 59, No. 6, 1311--1325 (1995; Zbl 0877.47026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 6, 207--222 (1995)] aimed to give a positive answer to a conjecture of Arveson [loc.\,cit.]. In Section~3, we consider in greater detail an improved and simplified construction of a nonstandard dynamical semigroup outlined in our short communication [\textit{A. S. Kholevo}, Russ. Math. Surv. 51, No. 6, 1206--1207 (1996; Zbl 1042.46513); translation from Usp. Mat. Nauk 51, No. 6, 225--226 (1996)].Fixed point theorems for self and non-self \(F\)-contractions in metric spaces endowed with a graph.https://www.zbmath.org/1456.470192021-04-16T16:22:00+00:00"Younus, Awais"https://www.zbmath.org/authors/?q=ai:younus.awais"Azam, Muhammad Umer"https://www.zbmath.org/authors/?q=ai:azam.muhammad-umer"Asif, Muhammad"https://www.zbmath.org/authors/?q=ai:asif.muhammad-salmanSummary: The main results obtained in this paper are fixed point theorems for self and non-self \(GF\)-contractions on metric spaces endowed with a graph. Our new results are generalization of recently fixed point theorems for self mappings on metric spaces and also fixed point theorems for non-self mappings in Banach spaces by using the concept of new type of contractive mappings namely \(F\)-contractions.Maximally entangled correlation sets.https://www.zbmath.org/1456.810422021-04-16T16:22:00+00:00"Alhajjar, Elie"https://www.zbmath.org/authors/?q=ai:alhajjar.elie"Russell, Travis B."https://www.zbmath.org/authors/?q=ai:russell.travis-bSummary: We study the set of quantum correlations generated by actions on maximally entangled states. We show that such correlations are dense in their own convex hull. As a consequence, we show that these correlations are dense in the set of synchronous quantum correlations. We introduce the concept of corners of correlation sets and show that every local or non-signalling correlation can be realized as the corner of a synchronous local or non-signalling correlation. We provide partial results for other correlation sets.Weighted composition operators from Dirichlet-type spaces into Stević-type spaces.https://www.zbmath.org/1456.300932021-04-16T16:22:00+00:00"Zhu, Xiangling"https://www.zbmath.org/authors/?q=ai:zhu.xianglingSummary: The boundedness and compactness of weighted composition operators from Dirichlet-type spaces into Stević-type spaces are investigated in this paper. Some estimates for the essential norm of weighted composition operators are also given.The phase space of one class of higher order Sobolev type equations in quasi-Banach spaces.https://www.zbmath.org/1456.350882021-04-16T16:22:00+00:00"Zamyshlyaeva, A. A."https://www.zbmath.org/authors/?q=ai:zamyshlyaeva.alena-aleksandrovna"Al Helli, Hamis M. A."https://www.zbmath.org/authors/?q=ai:al-helli.hamis-m-aSummary: The theory of Sobolev type equations experiences an epoch of blossoming. In this article the theory of higher order Sobolev type equations with relatively spectrally bounded operators, previously developed in Banach spaces, is transferred to quasi-Banach spaces. We use already well proved for solving Sobolev type equations phase space method, consisting in reduction of singular equation to regular one defined on some subspace of initial space. The propagators and the phase space of noncomplete higher order Sobolev type equations are constructed. Abstract results are illustrated by specific examples. The Boussinesq-Love equation in quasi-Banach space is considered as application.Analysis of traveling waveform of flexible waveguides containing absorbent material along flanged junctions.https://www.zbmath.org/1456.740852021-04-16T16:22:00+00:00"Afzal, Muhammad"https://www.zbmath.org/authors/?q=ai:afzal.muhammad-zeshan|afzal.muhammad-u"Shafique, Sajid"https://www.zbmath.org/authors/?q=ai:shafique.sajid"Wahab, Abdul"https://www.zbmath.org/authors/?q=ai:wahab.abdul-fatahSummary: The traveling waveform of a flexible waveguide bounded by elastic plates and with an inserted expansion chamber having flanges at two junctions and a finite elastic membrane atop is investigated through a mode-matching technique. The modeled problem is governed by Helmholtz's equation and includes Dirichlet and higher-order boundary conditions. An acoustically absorbent lining is placed along the inner sides of the flanges at the junctions while their outer sides are kept rigid. Moreover, the edge conditions are imposed to define the physical behavior of the elastic membrane and plates at finite edges. The configuration is excited with the structure as well as fluid-borne modes. The influence of the imposed edge conditions at the connections of the plates and the prescribed incident forcing on the transmission-loss along the duct is elaborated. Specifically, the effects of edge conditions on the transmission-loss of structure-borne vibrations and fluid-borne noise are specified. The performance of low-frequency approximation is compared with that of the benchmark mode-matching method and is found to be in good agreement with relative merits. Apposite numerical simulations are performed to substantiate the validity of the mode-matching technique.Invariant measures for systems of Kolmogorov equations.https://www.zbmath.org/1456.350312021-04-16T16:22:00+00:00"Addona, Davide"https://www.zbmath.org/authors/?q=ai:addona.davide"Angiuli, Luciana"https://www.zbmath.org/authors/?q=ai:angiuli.luciana"Lorenzi, Luca"https://www.zbmath.org/authors/?q=ai:lorenzi.lucaSummary: In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity.Dichotomy of iterated means for nonlinear operators.https://www.zbmath.org/1456.470202021-04-16T16:22:00+00:00"Saburov, Mansur"https://www.zbmath.org/authors/?q=ai:saburov.mansur-khSummary: In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains.Geometric ergodicity in a weighted Sobolev space.https://www.zbmath.org/1456.601742021-04-16T16:22:00+00:00"Devraj, Adithya"https://www.zbmath.org/authors/?q=ai:devraj.adithya"Kontoyiannis, Ioannis"https://www.zbmath.org/authors/?q=ai:kontoyiannis.ioannis"Meyn, Sean"https://www.zbmath.org/authors/?q=ai:meyn.sean-pSummary: For a discrete-time Markov chain \(\boldsymbol{X}=\{X(t)\}\) evolving on \(\mathbb{R}^{\ell}\) with transition kernel \(P\), natural, general conditions are developed under which the following are established:
\begin{itemize}
\item[(i)] The transition kernel \(P\) has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space \(L_{\infty}^{v,1}\) of functions with norm, \[ \Vert f\Vert_{v,1}=\mathop{\text{sup}}_{x\in\mathbb{R}^{\ell}}\frac{1}{v(x)}\max\{\vert f(x)\vert ,\vert \partial_1f(x)\vert ,\ldots,\vert \partial_{\ell}f(x)\vert \},\] where \(v\colon\mathbb{R}^{\ell}\to[1,\infty)\) is a Lyapunov function and \(\partial_i:=\partial/\partial x_i \).
\item[(ii)] The Markov chain is geometrically ergodic in \(L_{\infty}^{v,1}\): There is a unique invariant probability measure \(\pi\) and constants \(B<\infty\) and \(\delta>0\) such that, for each \(f\in L_{\infty}^{v,1}\), any initial condition \(X(0)=x\), and all \(t\geq0\): \begin{eqnarray*}\vert \mathsf{E}_x[f(X(t))]-\pi(f)\vert &\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\\\Vert \nabla\mathsf{E}_x[f(X(t))]\Vert_2&\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\end{eqnarray*} where \(\pi(f)=\int f\,d\pi \).
\item[(iii)] For any function \(f\in L_{\infty}^{v,1}\) there is a function \(h\in L_{\infty}^{v,1}\) solving Poisson's equation: \[h-Ph=f-\pi(f)\].
\end{itemize}
Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents. Relationships with topological coupling, in terms of the Wasserstein metric, are also explored.Invariant subspaces for commuting operators on a real Banach space.https://www.zbmath.org/1456.470032021-04-16T16:22:00+00:00"Lomonosov, V. I."https://www.zbmath.org/authors/?q=ai:lomonosov.victor-i"Shul'man, Viktor S."https://www.zbmath.org/authors/?q=ai:shulman.victor-sSummary: It is proved that the commutative algebra \(\mathcal A\) of operators on a reflexive real Banach space has an invariant subspace if each operator \(T\in\mathcal A\) satisfies the condition
\[
\| 1 - \varepsilon T^2\|_e \leqslant 1 + o(\varepsilon)\text{ as }\varepsilon \searrow 0,
\]
where \(\| \cdot\|_e\) denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.\(p\)-regularity and weights for operators between \(L^p\)-spaces.https://www.zbmath.org/1456.460262021-04-16T16:22:00+00:00"Sánchez Pérez, Enrique A."https://www.zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonso"Tradacete, Pedro"https://www.zbmath.org/authors/?q=ai:tradacete-perez.pedroSummary: We explore the connection between \(p\)-regular operators on Banach function spaces and weighted \(p\)-estimates. In particular, our results focus on the following problem. Given finite measure spaces \(\mu\) and \(\nu\), let \(T\) be an operator defined from a Banach function space \(X(\nu)\) and taking values on \(L^p (v d \mu)\) for \(v\) in certain family of weights \(V\subset L^1(\mu)_+\) we analyze the existence of a bounded family of weights \(W\subset L^1(\nu)_+\) such that for every \(v\in V\) there is \(w \in W\) in such a way that \(T:L^p(w d \nu) \to L^p(v d \mu)\) is continuous uniformly on \(V\). A condition for the existence of such a family is given in terms of \(p\)-regularity of the integration map associated to a certain vector measure induced by the operator \(T\).Geometric mean of partial positive definite matrices with missing entries.https://www.zbmath.org/1456.150302021-04-16T16:22:00+00:00"Choi, Hayoung"https://www.zbmath.org/authors/?q=ai:choi.hayoung"Kim, Sejong"https://www.zbmath.org/authors/?q=ai:kim.sejong"Shi, Yuanming"https://www.zbmath.org/authors/?q=ai:shi.yuanmingSummary: In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain properties which the weighted geometric mean of two positive definite matrices satisfies. Additionally, counterexamples demonstrate that certain properties do not hold. A Loewner order on partial Hermitian matrices is also defined. The known results for the maximum determinant positive completion are developed with an integral representation, and the results are applied to the weighted geometric mean of two partial positive definite matrices with missing entries. Moreover, a relationship between two positive definite completions is established with respect to their determinants, showing relationship between their entropy for a zero-mean, multivariate Gaussian distribution. Computational results as well as one application are shown.A norm inequality for positive block matrices.https://www.zbmath.org/1456.150212021-04-16T16:22:00+00:00"Lin, Minghua"https://www.zbmath.org/authors/?q=ai:lin.minghuaSummary: Any positive matrix \(M = (M_{i, j})_{i, j = 1}^m\) with each block \(M_{i, j}\) square satisfies the symmetric norm inequality \(\| M \| \leq \| \sum_{i = 1}^m M_{i, i} + \sum_{i = 1}^{m - 1} \omega_i I \|\), where \(\omega_i\) (\(i = 1, \ldots, m - 1\)) are quantities involving the width of numerical ranges. This extends the main theorem of \textit{J.-C. Bourin} and \textit{A. Mhanna} [C. R., Math., Acad. Sci. Paris 355, No. 10, 1077--1081 (2017; Zbl 06806461)]
to a higher number of blocks.Geometry of \(\ell_p^n\)-balls: classical results and recent developments.https://www.zbmath.org/1456.460132021-04-16T16:22:00+00:00"Prochno, Joscha"https://www.zbmath.org/authors/?q=ai:prochno.joscha"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christoph"Turchi, Nicola"https://www.zbmath.org/authors/?q=ai:turchi.nicolaSummary: In this article we first review some by-now classical results about the geometry of \(\ell_p\)-balls \(\mathbb{B}_p^n\) in \(\mathbb{R}^n\) and provide modern probabilistic arguments for them. We also present some more recent developments including a central limit theorem and a large deviations principle for the \(q\)-norm of a random point in \(\mathbb{B}_p^n\). We discuss their relation to the classical results and give hints to various extensions that are available in the existing literature.
For the entire collection see [Zbl 1431.60003].Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes.https://www.zbmath.org/1456.351512021-04-16T16:22:00+00:00"Biswas, Animikh"https://www.zbmath.org/authors/?q=ai:biswas.animikh"Hudson, Joshua"https://www.zbmath.org/authors/?q=ai:hudson.joshua"Tian, Jing"https://www.zbmath.org/authors/?q=ai:tian.jingSummary: In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces \(H^s, s > \frac{1}{2}\). This complements the seminal work of \textit{C. Foias} and \textit{R. Temam} on \(H^1\) based Gevrey classes [J. Funct. Anal. 87, No. 2, 359--369 (1989; Zbl 0702.35203)], thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for \(\frac{1}{2} < s < \frac{5}{2}\).Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex.https://www.zbmath.org/1456.170192021-04-16T16:22:00+00:00"Mukhamedov, Farrukh"https://www.zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Khakimov, Otabek"https://www.zbmath.org/authors/?q=ai:khakimov.otabek-n"Omirov, Bakhrom"https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-a"Qaralleh, Izzat"https://www.zbmath.org/authors/?q=ai:qaralleh.izzatAn algebra \((E,+,\cdot)\) over a field \(K\) is called an \textit{evolution algebra} provided it has a basis \(\{e_i\}\) such that \(e_i\cdot e_j=\) for everi \(i\neq j\) and \(e_i\cdot e_i=\sum_k a_{i,k}e_k\). Such a basis, is called \textit{natural basis}. This paper deals with nilpotent evolution algebras \(E\) such that \(\dim E^2= \dim E -1\).
The authors describe the Lie algebra of derivations of \(E\) when \(E\) has maximal index of nilpotency. Furthermore, they describe local and 2-local derivations for such algebras showing, for example, that every 2-local derivation of \(E\) is a derivation.
Finally, the authors determine the automorphisms and local automorphims for this type of algebras.
Reviewer: Antonio M. Oller Marcén (Zaragoza)Operators on weighted Lorentz-Karamata-Bochner spaces.https://www.zbmath.org/1456.470092021-04-16T16:22:00+00:00"Datt, Gopal"https://www.zbmath.org/authors/?q=ai:datt.gopal.1"Chugh, Renu"https://www.zbmath.org/authors/?q=ai:chugh.renu"Jakhar, Jagjeet"https://www.zbmath.org/authors/?q=ai:jakhar.jagjeetSummary: In this paper, conditions are derived for the multiplication, composition and weighted composition operators on weighted Lorentz-Karamata-Bochner (WLKB) spaces \(L_{p,q,w;b}\), \(1 < p\leq \infty\), \(1\leq q\leq\infty\), to be bounded.The joint numerical radius on \(C^*\)-algebras.https://www.zbmath.org/1456.460432021-04-16T16:22:00+00:00"Mabrouk, Mohamed"https://www.zbmath.org/authors/?q=ai:mabrouk.mohamedSummary: Let \(\mathfrak{A}\) be unital \(C^*\)-algebra with unit \(e\) and positive cone \(\mathfrak{A}^+\) such that every irreducible representation is infinite dimensional. For every \(\mathbf{a} =(a_1,\dots,a_n)\in\mathfrak{A}^n\), the joint numerical radius of \(\mathbf{a}\) is denoted by \(\mathbf{v}(\mathbf{a})\). It is shown that an element \(\mathbf{a}\in \mathfrak{A}^n\) satisfies \(\sum_{j=1}^n|f(a_j)|^2=1\) for every pure state \(f\) of \(\mathfrak{A}\) if and only if each \(a_j\) is in the center of \(\mathfrak{A}\) and \( \sum_{j=1}^na_j a_j^*=e\). Furthermore, we characterize elements \(\mathbf{a}_1, \dots, \mathbf{a}_n \in \mathfrak{A}^n\) such that for any \(\mathbf{x}\in (\mathfrak{A}^+)^n\) there exists \(\alpha =(\alpha_1, \dots, \alpha_n)\in \mathbb{R}^n\) such that \( \sum_{j=1}^{j=n}\alpha_j^2=1\) and \(\mathbf{v}(\sum_{j=1}^{j=n}\alpha_j\mathbf{a}_j+ \mathbf{x}) = 1+\mathbf{v}(\mathbf{x})\).
For the entire collection see [Zbl 1444.15003].The essential spectrum of the discrete Laplacian on Klaus-sparse graphs.https://www.zbmath.org/1456.051062021-04-16T16:22:00+00:00"Golénia, Sylvain"https://www.zbmath.org/authors/?q=ai:golenia.sylvain"Truc, Françoise"https://www.zbmath.org/authors/?q=ai:truc.francoiseSummary: \textit{M. Klaus} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 38, 7--13 (1983; Zbl 0527.47032)] studied a class of potentials with bumps and computed the essential spectrum of the associated Schrödinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the Appendix.Biduality in weighted spaces of analytic functions.https://www.zbmath.org/1456.460142021-04-16T16:22:00+00:00"Boyd, Christopher"https://www.zbmath.org/authors/?q=ai:boyd.christopher"Rueda, Pilar"https://www.zbmath.org/authors/?q=ai:rueda.pilarSummary: We study new conditions for non necessarily radial weights implying that the weighted Banach space \(\mathcal{H}_v(U)\) of analytic functions \(f\) such that \(vf\) is bounded on \(U\), is canonically isometrically isomorphic to the bidual of \(\mathcal{H}_{v_o}(U)\), its closed subspace formed by those functions \(f\) such that \(vf\) converges to \(0\) on the boundary of \(U\). We provide several examples of weights that satisfy these conditions. As an application, we show that whenever \(\mathcal{H}_v(U)=\mathcal{H}_{v_o}(U)''\) the norm-attaining functions are dense in \(\mathcal{H}_v(U)\).
For the entire collection see [Zbl 1444.15003].Projection extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces.https://www.zbmath.org/1456.650362021-04-16T16:22:00+00:00"Deng, Lanmei"https://www.zbmath.org/authors/?q=ai:deng.lanmei"Hu, Rong"https://www.zbmath.org/authors/?q=ai:hu.rong"Fang, Yaping"https://www.zbmath.org/authors/?q=ai:fang.yapingSummary: We present two projection extragradient algorithms for solving equilibrium problems without monotonicity and Lipschitz-type property in Hilbert spaces. Our strategy consists in embedding a subgradient projection step in the extragradient algorithm and employing an Armijo-linesearch. The strategy guarantees that the sequences generated by the presented algorithms converge weakly and strongly to a solution of the equilibrium problem, respectively. The convergence does not require any monotonicity and Lipschitz-type property of the bifunction but the nonemptyness of the solution set of the associated Minty equilibrium problem. Some numerical experiments illustrate the efficiency of the proposed algorithms.Strong convergence of approximants for Lipschitz pseudocontractive mappings.https://www.zbmath.org/1456.470342021-04-16T16:22:00+00:00"Qin, Xiaowei"https://www.zbmath.org/authors/?q=ai:qin.xiaowei"Wang, Ke"https://www.zbmath.org/authors/?q=ai:wang.ke.4|wang.ke.3|wang.ke.2|wang.ke.1|wang.ke"Chen, Rudong"https://www.zbmath.org/authors/?q=ai:chen.rudongSummary: The interest in pseudocontractive mappings is due mainly to their connection with nonlinear monotone operators. In the present paper, an iterative algorithm is introduced for approximating fixed points of Lipschitz pseudocontractive mappings. Strong convergence is demonstrated.Subgradient projection methods extended to monotone bilevel equilibrium problems in Hilbert spaces.https://www.zbmath.org/1456.650452021-04-16T16:22:00+00:00"Anh, Pham Ngoc"https://www.zbmath.org/authors/?q=ai:pham-ngoc-anh.|anh.pham-ngoc"Tu, Ho Phi"https://www.zbmath.org/authors/?q=ai:tu.ho-phiSummary: In this paper, by basing on the inexact subgradient and projection methods presented by \textit{P. Santos} and \textit{S. Scheimberg} [Comput. Appl. Math. 30, No. 1, 91--107 (2011; Zbl 1242.90265)], we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms.A bound on the pseudospectrum for a class of non-normal Schrödinger operators.https://www.zbmath.org/1456.350962021-04-16T16:22:00+00:00"Dondl, Patrick W."https://www.zbmath.org/authors/?q=ai:dondl.patrick-w"Dorey, Patrick"https://www.zbmath.org/authors/?q=ai:dorey.patrick-e"Rösler, Frank"https://www.zbmath.org/authors/?q=ai:rosler.frankSummary: We are concerned with the non-normal Schrödinger operator \(H = - \Delta + V\) on \(L^2(\mathbb{R}^n)\), where \(V \in W_{\operatorname{loc}}^{1, \infty}(\mathbb{R}^n)\) and \(\operatorname{Re} V(x) \geq c |x|^2 - d\) for some \(c, d > 0\). The spectrum of this operator is discrete and its real part is bounded below by \(- d\). In general, the \(\varepsilon\)-pseudospectrum of \(H\) will have an unbounded component for any \(\varepsilon > 0\) and thus will not approximate the spectrum in a global sense.
By exploiting the fact that the semigroup \(\operatorname{e}^{-tH}\) is immediately compact, we show a complementary result, namely that for every \(\delta > 0\), \(R > 0\) there exists an \(\varepsilon > 0\) such that the \(\varepsilon\)-pseudospectrum
\[
\sigma_\varepsilon(H) \subset \{z : \operatorname{Re} z \geq R \} \bigcup \bigcup_{\lambda \in \sigma(H)} \{z : |z - \lambda| < \delta \}.
\]
In particular, the unbounded part of the pseudospectrum escapes towards \(+ \infty\) as \(\varepsilon\) decreases. In addition, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.A fixed point approach to stability of \(k\)-th radical functional equation in non-Archimedean \((n,\beta)\)-Banach spaces.https://www.zbmath.org/1456.390032021-04-16T16:22:00+00:00"EL-Fassi, Iz-iddine"https://www.zbmath.org/authors/?q=ai:el-fassi.iz-iddine"Elqorachi, Elhoucien"https://www.zbmath.org/authors/?q=ai:elqorachi.elhoucien"Khodaei, Hamid"https://www.zbmath.org/authors/?q=ai:khodaei.hamidSummary: In this work, we prove a simple fixed point theorem in non-Archimedean \((n,\beta)\)-Banach spaces, by applying this fixed point theorem, we will study the stability and the hyperstability of the \(k\)th radical-type functional equation:
\[
f\left( \sqrt[k]{x^k+y^k}\right) = f(x)+f(y),
\]
where \(f\) is a mapping on the set of real numbers and \(k\) is a fixed positive integer. Furthermore, we give some important consequences from our main results.On compact operators on some sequence spaces related to matrix \(B(r,s,t)\).https://www.zbmath.org/1456.470102021-04-16T16:22:00+00:00"Demiriz, Serkan"https://www.zbmath.org/authors/?q=ai:demiriz.serkan"Kara, Emrah Evren"https://www.zbmath.org/authors/?q=ai:kara.emrah-evrenSummary: In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces \(c_0(B)\), \(\ell_\infty(B)\) and \(\ell_{p}(B)\) which have recently been introduced in [\textit{A. Sönmez}, Comput. Math. Appl. 62, No. 2, 641--650 (2011; Zbl 1228.40006)]. Further, by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces.Complete characterization of bounded composition operators on the general weighted Hilbert spaces of entire Dirichlet series.https://www.zbmath.org/1456.300032021-04-16T16:22:00+00:00"Doan, Minh Luan"https://www.zbmath.org/authors/?q=ai:doan.minh-luan"Lê, Hai Khôi"https://www.zbmath.org/authors/?q=ai:le.hai-khoiSummary: We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space being considered contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.Stable and non-symmetric pitchfork bifurcations.https://www.zbmath.org/1456.340382021-04-16T16:22:00+00:00"Pujals, Enrique"https://www.zbmath.org/authors/?q=ai:pujals.enrique-ramiro"Shub, Michael"https://www.zbmath.org/authors/?q=ai:shub.michael"Yang, Yun"https://www.zbmath.org/authors/?q=ai:yang.yunAuthors' abstract: In this paper, we present a criterion for pitchfork bifurcations of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.
Reviewer: Tao Li (Chengdu)Discussion on the existence of best proximity points in metric spaces.https://www.zbmath.org/1456.370282021-04-16T16:22:00+00:00"Hong, Shihuang"https://www.zbmath.org/authors/?q=ai:hong.shihuang"Zhou, Jie"https://www.zbmath.org/authors/?q=ai:zhou.jie.1|zhou.jie.4|zhou.jie.3|zhou.jie.2|zhou.jie"Chen, Ji"https://www.zbmath.org/authors/?q=ai:chen.ji"Hou, Haiyang"https://www.zbmath.org/authors/?q=ai:hou.haiyang"Wang, Li"https://www.zbmath.org/authors/?q=ai:wang.li.2|wang.li|wang.li.1|wang.li.3|wang.li.6|wang.li.4|wang.li.5Summary: In this paper, we modify the definition of some generalized proximal contractions and enumerate a list of equivalent conditions for various versions of generalized proximal contractions of non-self set-valued mappings on (ordered) metric spaces. By using the fixed point means, we establish the existence of best proximity points for mappings involving such contractions which extend and improve many existing related results, as well as, reveal that most of existing best proximity point theorems on metric spaces are in face equivalent and immediate consequences of well-known fixed point theorems. Finally, some examples are given to support our results.Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).The completion of generalized b-metric spaces and fixed points.https://www.zbmath.org/1456.540122021-04-16T16:22:00+00:00"Cobzaş, Ştefan"https://www.zbmath.org/authors/?q=ai:cobzas.stefan"Czerwik, Stefan"https://www.zbmath.org/authors/?q=ai:czerwik.stefanIn the article, the authors continue their research on the so-called b-metric spaces. Some fixed point theorems in b-metric spaces are proved.
Reviewer: Jarosław Górnicki (Rzeszów)A basis of \(\mathbb{R}^n\) with good isometric properties and some applications to denseness of norm attaining operators.https://www.zbmath.org/1456.460102021-04-16T16:22:00+00:00"Acosta, María D."https://www.zbmath.org/authors/?q=ai:acosta.maria-d"Dávila, José L."https://www.zbmath.org/authors/?q=ai:davila.jose-lThis paper deals with the Bishop-Phelps-Bollobás property introduced in [\textit{M. D. Acosta} et al., J. Funct. Anal. 254, No. 11, 2780--2799 (2008; Zbl 1152.46006)]. The Bishop-Phelps-Bollobás property of a pair \((X,Y)\) is an improved version of the denseness of the set of norm attaining (bounded linear) operators from \(X\) to \(Y\), in the same way that \textit{B. Bollobás} [Bull. Lond. Math. Soc., 2, 181--182 (1970; Zbl 0217.45104)] improved the classical Bishop-Phelps theorem [\textit{E. Bishop} and \textit{R. R. Phelps}, Bull. Am. Math. Soc., 67, 97--98 (1961; Zbl 0098.07905)]. A surprising result contained in the seminal paper [Acosta et al., loc. cit.] is that the Bishop-Phelps-Bollobás property may fail for a pair \((X,Y)\) even when the Banach space \(X\) is finite dimensional, despite the fact that, in this case, all operators from \(X\) to \(Y\) attain their norms. In that paper, the case when the domain space \(X\) is \(\ell_1^n\) (the \(n\)-dimensional \(\ell_1\) space) or even when \(X=\ell_1\), was studied and the Bishop-Phelps-Bollobás property was characterized in terms of a property of \(Y\) called the approximate hyperplane series property (AHSP), which deals with convex series in the space \(Y\). On the other hand, it is known that when \(X\) is a uniformly convex Banach space, then every pair \((X,Y)\) has the Bishop-Phelps-Bollobás property, regardless of the range space \(Y\), see [\textit{S. K. Kim} and \textit{H. J. Lee}, Can. J. Math. 66, No. 2, 373--386 (2014; Zbl 1298.46016)] or [\textit{M. D. Acosta} et al., Trans. Am. Math. Soc. 365, No. 11, 5911--5932 (2013; Zbl 1296.46007)].
In the paper under review, the case when \(X\) is \(\ell_\infty^n\) (the \(n\)-dimensional \(\ell_\infty\) space) is studied. The authors characterize those Banach spaces \(Y\) for which the pair \((\ell_\infty^n,Y)\) has the Bishop-Phelps-Bollobás property in terms of the behaviour of some subset of \(Y^n\). To get their results, the authors introduce and study a particular vector basis of \(\mathbb{R}^n\).
As the main application of their results, the authors show that the pairs \((\ell_\infty^n,L_1(\mu))\) have the Bishop-Phelps-Bollobás property for every positive measure \(\mu\) and every positive integer~\(n\).
Reviewer: Miguel Martín (Granada)New approximation schemes for two asymptotically perturbed nonexpansive nonself mappings.https://www.zbmath.org/1456.470292021-04-16T16:22:00+00:00"Piwma, Nattanawan"https://www.zbmath.org/authors/?q=ai:piwma.nattanawan"Thianwan, Tanakit"https://www.zbmath.org/authors/?q=ai:thianwan.tanakitSummary: In this paper, we introduce and study a new type of two-step iterative scheme for two asymptotically perturbed nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for the new two-step iterative scheme in a uniformly convex Banach space. The results obtained in this paper generalize and refine some known results in the current literature.Weighted boundedness of the fractional maximal operator and Riesz potential generated by Gegenbauer differential operator.https://www.zbmath.org/1456.420202021-04-16T16:22:00+00:00"Ibrahimov, Elman J."https://www.zbmath.org/authors/?q=ai:ibrahimov.elman-j"Guliyev, Vagif S."https://www.zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Jafarova, Saadat A."https://www.zbmath.org/authors/?q=ai:jafarova.saadat-aSummary: In the paper we study the weighted \((L_{p,\omega,\lambda}, L_{q,\omega,\lambda}) \)-boundedness of the fractional maximal operator \(M_G^\alpha \) (\(G\) is a fractional maximal operator) and the Riesz potential (\(G\) is the Riesz potential) generated by the Gegenbauer differential operator
\[G_{\lambda}\equiv G=( x^{2}-1) ^{\frac{1}{2}-\lambda}\frac{d}{dx}( x^{2}-1) ^{\lambda +\frac{1}{2}}\frac{d}{dx}, \quad x\in (1,\infty ),\quad \lambda \in \left( 0,\frac{1}{2}\right).\]A new low-cost double projection method for solving variational inequalities.https://www.zbmath.org/1456.651902021-04-16T16:22:00+00:00"Gibali, Aviv"https://www.zbmath.org/authors/?q=ai:gibali.aviv"Thong, Duong Viet"https://www.zbmath.org/authors/?q=ai:duong-viet-thong.Summary: In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of [\textit{Y. Censor} et al., J. Optim. Theory Appl. 148, No. 2, 318--335 (2011; Zbl 1229.58018)] and Popov's method. The proposed scheme combines some of the advantages of the methods mentioned above, first it requires only one orthogonal projection onto the feasible set of the problem while the next computation has a closed formula. Second, only one mapping evaluation is required per each iteration and there is also a usage of an adaptive step size rule that avoids the need to know the Lipschitz constant of the associated mapping. We present two convergence theorems of the proposed method, weak convergence result which requires pseudomonotonicity, Lipschitz and sequentially weakly continuity of the associated mapping and strong convergence theorem with rate of convergence which requires Lipschitz continuity and strongly pseudomonotone only. Primary numerical experiments and comparisons demonstrate the advantages and potential applicability of the new scheme.Continuity of composition operators in Sobolev spaces.https://www.zbmath.org/1456.460282021-04-16T16:22:00+00:00"Bourdaud, Gérard"https://www.zbmath.org/authors/?q=ai:bourdaud.gerard"Moussai, Madani"https://www.zbmath.org/authors/?q=ai:moussai.madaniGiven a function \(f:\mathbb{R}\to\mathbb{R}\), the composition operator \(T_f\) is defined by \(T_f(g):= f\circ g\), where \(g\) runs over some function space. The problem of ``automatic continuity'' of \(T_f\) consists in verifying or falsifying that \(T_f\) is always continuous in a function space \(X\) whenever \(T_f(X)\subseteq X\). In some cases (e.g., for \(X= C\)) the answer to this problem is completely trivial, in other cases surprisingly difficult, in still other cases simply unknown. For instance, the (positive) answer was given for \(X= C^1\) by \textit{J. Appell} et al. [Boll. Unione Mat. Ital. (9) 4, No. 3, 321--336 (2011; Zbl 1229.47086)], while the (negative) answer for \(X=\text{Lip}\) by means of a counterexample due to \textit{M. Z. Berkolaiko} [in: Trudy Sem. funkcional. Analizu 12, Voronez, 96--104 (1969; Zbl 0266.47053)]. Remarkably, in the case \(X=BV\) the problem of automatic continuity remained open for many decades, and was solved (positively) only recently by \textit{P. Maćkowiak} [Aequationes Math. 91, No. 4, 759--777 (2017; Zbl 06787181)] and, with a much simpler and elegant proof, by \textit{S. Reinwand} [Real Anal. Exch. 45, No. 1, 173--204 (2020; Zbl 1440.26004)].
The problem of automatic continuity is particularly delicate when \(T_f\) is considered in the Sobolev space \(X= W^m_p(\mathbb{R}^n)\) for \(1\le p<\infty\), \(m\in\mathbb{N}_0\), and \(n\in\mathbb{N}\). The answer is positive in many cases, as was shown by \textit{A. Ancona} for \(m=1\) and \(p=2\) [C. R. Acad. Sci., Paris, Sér. A 282, 871--873 (1976; Zbl 0322.31011)], by \textit{M. Marcus} and \textit{V. J. Mizel} [J. Funct. Anal. 33, 217--229 (1979; Zbl 0418.46024)] for \(m=1\) and \(p>1\), by \textit{G. Bourdaud} and \textit{M. Lanza de Cristoforis} [Stud. Math. 184, No. 3, 271--298 (2008; Zbl 1139.46030)] for \(m> n/p\), \(m\ge 2\) and \(p>1\), and by \textit{B. E. J. Dahlberg} [Proc. Symp. Pure Math. 35, 183--185 (1979; Zbl 0421.46027)] for \(1+ 1/p<m< n/p\), where the last case leads to a strong degeneracy.
Summarizing, so far the three cases \(X= W^2_1(\mathbb{R}^n)\) with \(n\ge 3\), \(X= W^{n/p}_p(\mathbb{R}^n)\) with \(n>p>1\), and \(X= W^m_1(\mathbb{R}^n)\) with \(m\ge\max\{n, 2\}\) remained open, and the aim of the authors of the present paper is to fill this gap. To this end, they reduce all these cases to the problem of automatic continuity in the special potential space \(X= W^m_p(\mathbb{R}^n)\cap \dot{W}^1_{mp}(\mathbb{R}^n)\) considered previously by \textit{D. R. Adams} and \textit{M. Frazier} [Proc. Am. Math. Soc. 114, No. 1, 155--165 (1992; Zbl 0753.47045)]; in these spaces the problem is completely solved. Thus, this paper may be considered as a milestone in the theory of the composition operator which, in spite of its simple form, exhibits many surprising and unexpected phenomena.
Reviewer: Jürgen Appell (Würzburg)Some coincidence and common fixed point results in cone metric spaces over Banach algebras via weak \(g\)-\(\varphi \)-contractions.https://www.zbmath.org/1456.540142021-04-16T16:22:00+00:00"Malhotra, S. K."https://www.zbmath.org/authors/?q=ai:malhotra.sandeep-kumar"Bhargava, P. K."https://www.zbmath.org/authors/?q=ai:bhargava.pradeep-kumar"Shukla, S."https://www.zbmath.org/authors/?q=ai:shukla.satishSummary: Recently, \textit{B. Li} and \textit{H. Huang} [J. Funct. Spaces 2017, Article ID 5054603, 6 p. (2017; Zbl 1456.54013)] introduced the notion of weak \(\varphi \)-contractions on cone metric spaces over Banach algebras. The purpose of this paper is to generalize the main result of Li and Huang [loc. cit.] by proving some coincidence and common fixed point results in cone metric spaces over Banach algebras via weak \(g\)-\( \varphi \)-contractions for weakly compatible mappings. Some examples are presented which verify and illustrate the results proved herein.On Meir-Keeler contraction in Branciari \(b\)-metric spaces.https://www.zbmath.org/1456.540152021-04-16T16:22:00+00:00"Mitrović, Z."https://www.zbmath.org/authors/?q=ai:mitrovic.zoran-d"Radenović, S."https://www.zbmath.org/authors/?q=ai:radenovic.stojanSummary: In this paper we consider Meir-Keeler type results in the context of Branciari \(b\)-metric spaces. Our results generalize, improve and complement several ones in the existing literature.Sharp bounds for oscillatory integral operators with homogeneous polynomial phases.https://www.zbmath.org/1456.420162021-04-16T16:22:00+00:00"He, Danqing"https://www.zbmath.org/authors/?q=ai:he.danqing"Shi, Zuoshunhua"https://www.zbmath.org/authors/?q=ai:shi.zuoshunhuaSummary: We obtain sharp \(L^p\) bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition that is an important notion introduced by \textit{A. Greenleaf} et al. [J. Funct. Anal. 244, No. 2, 444--487 (2007; Zbl 1127.35090)]. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint \(L^p\) estimates.Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition.https://www.zbmath.org/1456.340142021-04-16T16:22:00+00:00"Bondarenko, Natalia P."https://www.zbmath.org/authors/?q=ai:bondarenko.natalia-pThe paper deals with the boundary value problem
\[
-y''(x)+q(x)y(x)=\lambda y(x),\; 0<x<\pi,
\]
\[
y(0)=0,\; f_1(\lambda)y'(\pi)+f_2(\lambda)y(\pi)=0,
\]
where \(f_k(\lambda)\) are entire functions in \(\lambda.\) The author studies the inverse problem of recovering the potential \(q(x)\) from a part of the spectrum. Local and global solvability are established for the solution of this nonlinear inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)The regularity of inverses to Sobolev mappings and the theory of \(\mathcal{Q}_{q,p} \)-homeomorphisms.https://www.zbmath.org/1456.300462021-04-16T16:22:00+00:00"Vodopyanov, S. K."https://www.zbmath.org/authors/?q=ai:vodopyanov.serguei-kSummary: We prove that each homeomorphism \(\varphi:D\to D^{\prime}\) of Euclidean domains in \(\mathbb{R}^n\), \(n\geq 2 \), belonging to the Sobolev class \(W^1_{p,\operatorname{loc}}(D) \), where \(p\in[1,\infty) \), and having finite distortion induces a bounded composition operator from the weighted Sobolev space \(L^1_p(D^{\prime};\omega)\) into \(L^1_p(D)\) for some weight function \(\omega:D^{\prime}\to(0,\infty) \). This implies that in the cases \(p>n-1\) and \(n\geq 3\) as well as \(p\geq 1\) and \(n\geq 2\) the inverse \(\varphi^{-1}:D^{\prime}\to D\) belongs to the Sobolev class \(W^1_{1,\operatorname{loc}}(D^{\prime}) \), has finite distortion, and is differentiable \({\mathcal{H}}^n \)-almost everywhere in \(D^{\prime} \). We apply this result to \(\mathcal{Q}_{q,p} \)-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of \(\mathcal{Q}_{q,p} \)-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.On reverses of the Golden-Thompson type inequalities.https://www.zbmath.org/1456.150202021-04-16T16:22:00+00:00"Ghaemi, Mohammad Bagher"https://www.zbmath.org/authors/?q=ai:ghaemi.mohammad-bagher"Kaleibary, Venus"https://www.zbmath.org/authors/?q=ai:kaleibary.venus"Furuichi, Shigeru"https://www.zbmath.org/authors/?q=ai:furuichi.shigeruSummary: In this paper we present some reverses of the Golden-Thompson type inequalities:
Let $H$ and $K$ be Hermitian matrices such that $ e^s e^H \preceq_{ols} e^K \preceq_{ols} e^t e^H$ for
some scalars $s \leq t$, and $\alpha \in [0 , 1]$. Then for all $p>0$ and $k =1,2,\ldots, n$
\[
\lambda_k (e^{(1-\alpha)H + \alpha K} ) \leq (\max \lbrace S(e^{sp}), S(e^{tp})\rbrace)^{\frac{1}{p}} \lambda_k (e^{pH} \sharp_\alpha e^{pK})^{\frac{1}{p}},
\]
where $A\sharp_\alpha B = A^\frac{1}{2} \big ( A^{-\frac{1}{2}} B^\frac{1}{2} A^{-\frac{1}{2}} \big) ^\alpha A^\frac{1}{2}$ is $\alpha$-geometric mean, $S(t)$ is the so called Specht's ratio and $\preceq_{ols}$ is the so called Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results.Spectral analysis of the Laplacian acting on discrete cusps and funnels.https://www.zbmath.org/1456.811752021-04-16T16:22:00+00:00"Athmouni, Nassim"https://www.zbmath.org/authors/?q=ai:athmouni.nassim"Ennaceur, Marwa"https://www.zbmath.org/authors/?q=ai:ennaceur.marwa"Golénia, Sylvain"https://www.zbmath.org/authors/?q=ai:golenia.sylvainSummary: We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique.1-d quantum harmonic oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms.https://www.zbmath.org/1456.351752021-04-16T16:22:00+00:00"Liang, Zhenguo"https://www.zbmath.org/authors/?q=ai:liang.zhenguo"Zhao, Zhiyan"https://www.zbmath.org/authors/?q=ai:zhao.zhiyan.1|zhao.zhiyan"Zhou, Qi"https://www.zbmath.org/authors/?q=ai:zhou.qi|zhou.qi.1|zhou.qi.2Summary: For a family of 1-d quantum harmonic oscillators with a perturbation which is \(C^2\) parametrized by \(E\in\mathcal{I}\subset\mathbb{R}\) and quadratic on \(x\) and \(-\operatorname{i} \partial_x\) with coefficients quasi-periodically depending on time \(t\), we show the reducibility (i.e., conjugation to time-independent) for a.e. \(E\). As an application of reducibility, we describe the behaviors of solutions in Sobolev space:
\begin{itemize}
\item Boundedness w.r.t. \(t\) is always true for ``most'' \(E\in\mathcal{I}\).
\item For ``generic'' time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. \(t\) occur for \(E\) in a ``small'' part of \(\mathcal{I}\).
\end{itemize}
Concrete examples are given for which the growths of Sobolev norm do occur.Book review of: B. Host and B. Kra, Nilpotent structures in ergodic theory.https://www.zbmath.org/1456.000122021-04-16T16:22:00+00:00"Frantzikinakis, Nikos"https://www.zbmath.org/authors/?q=ai:frantzikinakis.nikosReview of [Zbl 1433.37001].Fractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions.https://www.zbmath.org/1456.340072021-04-16T16:22:00+00:00"Subramanian, Muthaiah"https://www.zbmath.org/authors/?q=ai:subramanian.muthaiah"Manigandan, Murugesan"https://www.zbmath.org/authors/?q=ai:manigandan.murugesan"Gopal, Thangaraj Nandha"https://www.zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.Properties of some breather solutions of a nonlocal discrete NLS equation.https://www.zbmath.org/1456.370812021-04-16T16:22:00+00:00"Ben, Roberto I."https://www.zbmath.org/authors/?q=ai:ben.roberto-i"Borgna, Juan Pablo"https://www.zbmath.org/authors/?q=ai:borgna.juan-pablo"Panayotaros, Panayotis"https://www.zbmath.org/authors/?q=ai:panayotaros.panayotisSummary: We present results on breather solutions of a discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by \textit{R. I. Ben} et al. [Phys. Lett., A 379, No. 30--31, 1705--1714 (2015; Zbl 1343.35210)] showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work, we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in [\textit{R. I. Ben} et al., Phys. Lett., A 379, No. 30--31, 1705--1714 (2015; Zbl 1343.35210)]. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.An inverse function theorem converse.https://www.zbmath.org/1456.460672021-04-16T16:22:00+00:00"Lawson, Jimmie"https://www.zbmath.org/authors/?q=ai:lawson.jimmie-dSummary: We establish the following converse of the well-known inverse function theorem. Let \(g : U \to V\) and \(f : V \to U\) be inverse homeomorphisms between open subsets of Banach spaces. If \(g\) is differentiable of class \(C^p\) and \(f\) is locally Lipschitz, then the Fréchet derivative of \(g\) at each point of \(U\) is invertible and \(f\) must be differentiable of class \(C^p\).Geometric matrix midranges.https://www.zbmath.org/1456.150312021-04-16T16:22:00+00:00"Mostajeran, Cyrus"https://www.zbmath.org/authors/?q=ai:mostajeran.cyrus"Grussler, Christian"https://www.zbmath.org/authors/?q=ai:grussler.christian"Sepulchre, Rodolphe"https://www.zbmath.org/authors/?q=ai:sepulchre.rodolphe-jA new projection-type method for solving multi-valued mixed variational inequalities without monotonicity.https://www.zbmath.org/1456.490192021-04-16T16:22:00+00:00"Wang, Zhong-bao"https://www.zbmath.org/authors/?q=ai:wang.zhongbao"Chen, Zhang-you"https://www.zbmath.org/authors/?q=ai:chen.zhangyou"Xiao, Yi-bin"https://www.zbmath.org/authors/?q=ai:xiao.yibin"Zhang, Cong"https://www.zbmath.org/authors/?q=ai:zhang.congSummary: In this paper, a new projection-type algorithm for solving multi-valued mixed variational inequalities without monotonicity is presented. Under some suitable assumptions, it is showed that the sequence generated by the proposed algorithm converges globally to a solution of the multi-valued mixed variational inequality considered. The algorithm exploited in this paper is based on the generalized \(f\)-projection operator due to \textit{K. Wu} and \textit{N. Huang} [Bull. Aust. Math. Soc. 73, No. 2, 307--317 (2006; Zbl 1104.47053)] rather than the well-known resolvent operator. Preliminary computational experience is also reported. The results presented in this paper generalize and improve some known results given in the literature.Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices.https://www.zbmath.org/1456.150052021-04-16T16:22:00+00:00"Dai, Hui"https://www.zbmath.org/authors/?q=ai:dai.hui"Geary, Zachary"https://www.zbmath.org/authors/?q=ai:geary.zachary"Kadanoff, Leo P."https://www.zbmath.org/authors/?q=ai:kadanoff.leo-pOn extrapolation properties of Schatten-von Neumann classes.https://www.zbmath.org/1456.470052021-04-16T16:22:00+00:00"Lykov, K. V."https://www.zbmath.org/authors/?q=ai:lykov.konstantin-vSummary: For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.The regularization method for solving variational inclusion problems.https://www.zbmath.org/1456.470282021-04-16T16:22:00+00:00"Pholasa, Nattawut"https://www.zbmath.org/authors/?q=ai:pholasa.nattawut"Cholamjiak, Prasit"https://www.zbmath.org/authors/?q=ai:cholamjiak.prasitSummary: In this paper, we introduce a new regularization method for solving inclusion problems in Banach spaces. We then prove strong convergence theorems under some mild conditions. Finally, we also provide some concrete examples including numerical experiments.An operational construction of the sum of two non-commuting observables in quantum theory and related constructions.https://www.zbmath.org/1456.810102021-04-16T16:22:00+00:00"Drago, Nicolò"https://www.zbmath.org/authors/?q=ai:drago.nicolo"Mazzucchi, Sonia"https://www.zbmath.org/authors/?q=ai:mazzucchi.sonia"Moretti, Valter"https://www.zbmath.org/authors/?q=ai:moretti.valterSummary: The existence of a real linear space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables \(A, B\) is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form \(aA+bB\) (\(a,b\in\mathbb{R}\)) if such measuring instruments are given for the addends observables \(A\) and \(B\) when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of \(f(aA+bB)\) out of the spectral measures of \(A\) and \(B\). We present such a construction with a formula which is valid for general unbounded self-adjoint operators \(A\) and \(B\), whose spectral measures may not commute, and a wide class of functions \(f: \mathbb{R}\rightarrow\mathbb{C} \). In the bounded case, we prove that the Jordan product of \(A\) and \(B\) (and suitably symmetrized polynomials of \(A\) and \(B)\) can be constructed with the same procedure out of the spectral measures of \(A\) and \(B\). The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.Domain wall partition functions and KP.https://www.zbmath.org/1456.822582021-04-16T16:22:00+00:00"Foda, O."https://www.zbmath.org/authors/?q=ai:foda.omar"Wheeler, M."https://www.zbmath.org/authors/?q=ai:wheeler.michael"Zuparic, M."https://www.zbmath.org/authors/?q=ai:zuparic.mathew-lCertain \(\ast\)-homomorphisms acting on unital \(C^\ast\)-probability spaces and semicircular elements induced by \(p\)-adic number fields over primes \(p\).https://www.zbmath.org/1456.460532021-04-16T16:22:00+00:00"Cho, Ilwoo"https://www.zbmath.org/authors/?q=ai:cho.ilwooSummary: In this paper, we study the Banach \(*\)-probability space
\((A\otimes_{\mathbb{C}}\mathbb{LS}, \tau_A^0)\) generated by a fixed unital \(C^*\)-probability space \((A, \varphi_A)\), and the semicircular elements \(\Theta_{p,j}\) induced by \(p\)-adic number fields \(\mathbb{Q}_p\), for all \(p \in \mathcal{P}\), \(j\in\mathbb{Z}\), where \(\mathcal{P}\) is the set of all primes, and \(\mathbb{Z}\) is the set of all integers. In particular, from the order-preserving shifts \(g\times h_\pm\) on \(\mathcal{P} \times \mathbb{Z}\), and \(*\)-homomorphisms \(\theta\) on \(A\), we define the corresponding \(*\)-homomorphisms \(\sigma_{(\pm ,1)}^{1:\theta}\) on \(A\otimes_{\mathbb{C}} \mathbb{LS}\), and consider free-distributional data affected by them.Dimensional reduction and scattering formulation for even topological invariants.https://www.zbmath.org/1456.814942021-04-16T16:22:00+00:00"Schulz-Baldes, Hermann"https://www.zbmath.org/authors/?q=ai:schulz-baldes.hermann"Toniolo, Daniele"https://www.zbmath.org/authors/?q=ai:toniolo.danieleSummary: Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the \(K\)-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.\(H^p\)-boundedness of Hankel Hausdorff operator involving Hankel transformation.https://www.zbmath.org/1456.420242021-04-16T16:22:00+00:00"Upadhyay, S. K."https://www.zbmath.org/authors/?q=ai:upadhyay.santosh-kumar"Pandey, Ravi Shankar"https://www.zbmath.org/authors/?q=ai:pandey.ravi-shankar"Mohapatra, R. N."https://www.zbmath.org/authors/?q=ai:mohapatra.ram-narayan|mohapatra.rabindra-nSummary: In this paper the \(H^p(0,\infty)\)-boundedness of Hankel Hausdorff operator involving Hankel transformation is investigated by the method used by \textit{Y. Kanjin} [Stud. Math. 148, No. 1, 37--45 (2001; Zbl 1001.47018)]. Some properties related to Hankel Hausdorff operator on \(H^p(0,\infty)\) space are discussed.Volterra integral operators from \(\mathcal{D}^p_{p-2+s}\) into \(F(p\lambda,p\lambda+s\lambda-2,q)\).https://www.zbmath.org/1456.300952021-04-16T16:22:00+00:00"Shen, Conghui"https://www.zbmath.org/authors/?q=ai:shen.conghui"Lou, Zengjian"https://www.zbmath.org/authors/?q=ai:lou.zengjian"Li, Songxiao"https://www.zbmath.org/authors/?q=ai:li.songxiaoSummary: Let \(1<p<\infty\), \(0<q<\infty\), \(0<s\), \(\lambda\leqslant 1\) such that \(q+s\lambda>1\). We characterize the boundedness and compactness of inclusion mapping from Dirichlet type spaces \(\mathscr{D}^p_{p-2+s}\) into tent spaces \(T_{p\lambda,q}(\mu)\). As an application, the boundedness of the Volterra operator \(T_g\), its companion operator \(I_g\) and the multiplication operator \(M_g\) from \(\mathscr{D}^p_{p-2+s}\) to \(F(p\lambda,p\lambda+s\lambda-2,q)\) are given. Furthermore, we study the essential norm and compactness of \(T_g\) and \(I_g\).Complexifications of real Banach spaces and their isometries.https://www.zbmath.org/1456.460122021-04-16T16:22:00+00:00"Ilišević, Dijana"https://www.zbmath.org/authors/?q=ai:ilisevic.dijana"Kuzma, Bojan"https://www.zbmath.org/authors/?q=ai:kuzma.bojan"Li, Chi-Kwong"https://www.zbmath.org/authors/?q=ai:li.chi-kwong"Poon, Edward"https://www.zbmath.org/authors/?q=ai:poon.edwardLet \((\mathcal{X}, \|\cdot\|)\) be a real normed space and \(\mathbb{C}\mathcal{X}:=\mathcal{X}+i\mathcal{X}\) be its complexification endowed with the so-called Taylor complexification norm given by \[\|x+iy\|_{\mathbb{C}}:=\sup_{0\le\theta\le 2\pi}\|x\cos\theta+y\sin\theta\|.\]
When \(\mathcal{X}\) is finite-dimensional, the authors determine the group of isometries on the space \((\mathbb{C} \mathcal{X},\|\cdot\|_{\mathbb{C}})\) in terms of those of \((\mathcal{X}, \|\cdot\|)\). The proof is very technical and uses a series of auxiliary lemmas and results that are of independent interest. Among them, the authors give a description of the Taylor complexification norm in terms of extreme points of the unit ball for the dual of the original norm \(\|\cdot\|\). As an application, they compute the group of isometries for the numerical radius and related norms. Other results, including a partial extension to infinite-dimensional Banach spaces, are also discussed. Moreover, many examples and remarks are given which nicely illustrate the obtained results.
Reviewer: Abdellatif Bourhim (Syracuse)Solutions for a singular Hadamard-type fractional differential equation by the spectral construct analysis.https://www.zbmath.org/1456.340082021-04-16T16:22:00+00:00"Zhang, Xinguang"https://www.zbmath.org/authors/?q=ai:zhang.xinguang"Yu, Lixin"https://www.zbmath.org/authors/?q=ai:yu.lixin"Jiang, Jiqiang"https://www.zbmath.org/authors/?q=ai:jiang.jiqiang"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.Remarks on the nonlocal Dirichlet problem.https://www.zbmath.org/1456.350592021-04-16T16:22:00+00:00"Grzywny, Tomasz"https://www.zbmath.org/authors/?q=ai:grzywny.tomasz"Kassmann, Moritz"https://www.zbmath.org/authors/?q=ai:kassmann.moritz"Leżaj, Łukasz"https://www.zbmath.org/authors/?q=ai:lezaj.lukaszSummary: We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.Fixed point theorems for generalized semi-quasi contractions.https://www.zbmath.org/1456.540162021-04-16T16:22:00+00:00"Pant, Rajendra"https://www.zbmath.org/authors/?q=ai:pant.rajendraSummary: In this note, we introduce the notion of a generalized semi-quasi contraction and obtain a fixed point theorem for such contraction. Our results extend and generalize some well-known fixed point theorems including Ćirić's quasi contraction theorem. As an application of our main theorem the existence of a solution for a class of functional equations arising in dynamic programming is discussed. At the end an open problem is also posed.Semigroup maximal functions, Riesz transforms, and Morrey spaces associated with Schrödinger operators on the Heisenberg groups.https://www.zbmath.org/1456.420252021-04-16T16:22:00+00:00"Wang, Hua"https://www.zbmath.org/authors/?q=ai:wang.hua|wang.hua.1|wang.hua.2Summary: Let \(\mathcal{L}=- \Delta_{\mathbb{H}^n}+V\) be a Schrödinger operator on the Heisenberg group \(\mathbb{H}^n\), where \(\Delta_{\mathbb{H}^n}\) is the sub-Laplacian on \(\mathbb{H}^n\) and the nonnegative potential \(V\) belongs to the reverse Hölder class \(\mathcal{B}_q\) with \(q\in [Q/2,\infty)\). Here, \(Q=2n+2\) is the homogeneous dimension of \(\mathbb{H}^n\). Assume that \(\{ e^{- t \mathcal{L}}\}_{t>0}\) is the heat semigroup generated by \(\mathcal{L}\). The semigroup maximal function related to the Schrödinger operator \(\mathcal{L}\) is defined by \(\mathcal{T}_{\mathcal{L}}^\ast (f)(u) := \sup_{t>0} | e^{- t \mathcal{L}} f (u)|\). The Riesz transform associated with the operator \(\mathcal{L}\) is defined by \(\mathcal{R}_{\mathcal{L}}= \nabla_{\mathbb{H}^n} \mathcal{L}^{-1/2}\), and the dual Riesz transform is defined by \(\mathcal{R}_{\mathcal{L}}^\ast= \mathcal{L}^{-1/2} \nabla_{\mathbb{H}^n} \), where \(\nabla_{\mathbb{H}^n}\) is the gradient operator on \(\mathbb{H}^n\). In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator \(\mathcal{L}\) on \(\mathbb{H}^n\). Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators \(\mathcal{T}_{\mathcal{L}}^\ast\), \(\mathcal{R}_{\mathcal{L}}\), and \(\mathcal{R}_{\mathcal{L}}^\ast\) acting on the Morrey spaces. In addition, it is shown that the Riesz transform \(\mathcal{R}_{\mathcal{L}}= \nabla_{\mathbb{H}^n} \mathcal{L}^{-1/2}\) is of weak-type \((1,1)\). It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.A course in analysis. Vol. V: Functional analysis, some operator theory, theory of distributions.https://www.zbmath.org/1456.460012021-04-16T16:22:00+00:00"Jacob, Niels"https://www.zbmath.org/authors/?q=ai:jacob.niels"Evans, Kristian P."https://www.zbmath.org/authors/?q=ai:evans.kristian-pThis fifth instalment of the multi-volume Course in Analysis (the previous parts are [Zbl 1327.26012], [Zbl 1401.26001], [Zbl 1381.28002] and [Zbl 1407.00004]) is dedicated to three main themes: functional analysis, (a non-comprehensive presentation of) operator theory, and distributions, continuing in an organic manner its preceding parts. The high standards of this series are successfully maintained, their main attributes (clarity of the presentation, perfectly adapted depth of the material for a wide readership, excellently chosen problems for illustrating the presented theory, etc.) being easily recognizable in the current volume as well. The rich experience of the authors in teaching analysis (in the broadest sense) transpires on every page and I highly recommend this book to anyone teaching or studying the topics presented.
The more than 850-pages long volume is divided into three parts dedicated to the main themes mentioned above, containing also a preface, an introduction, a list of symbols as well as three appendices, a symbol index, a continuation of the list of contributors to analysis from the first four volumes as well as solutions to the proposed problems and the standard comprehensive list of references.
In the first (and main, given its topic and length) part of the book the reader takes a 14-stop journey (mimicking maybe the 14~chapters of the part dedicated to Fourier Analysis in the previous volume of this series) in the world of Functional Analysis, dealing with topics such as infinite-dimensional vector spaces (in particular, topological vector spaces, Banach and Hilbert spaces as well as dual spaces), linear functionals, linear, adjoint and unbounded operators, Fredholm theory, spectral theory (including the Gelfand-Naimark theory) and self-adjoint operators, convexity and integral representations as well as various themes grouped under the headline ``Selected topics''.
The second part contains altogether seven~chapters on, as the authors put it, ``some'' operator theory, where integral operators, one-parameter semigroups of operators, positivity preserving operators and Markovian semigroups, regular Sturm-Liouville problems, a brief introduction to Sobolev spaces, operators induced by the Dirichlet problem, and, again, some selected topics are presented.
This is followed by a six-chapter third part on distribution theory, dealing with subjects such as function spaces (in particular Fréchet spaces), distributions in the sense of Schwartz, tempered distributions and the Fourier transform, tensor products, kernels, and Calderón-Zygmund operators. The three appendices are dedicated to completeness, nets and the Riesz representation theorem.
Like the previous volumes of the series, the present one is logically constructed and, by paying great attention to details (such as suitable examples for justifying generalizations, remarks stressing surprising connections and properties, suitable problems for illustrating the theory, etc.), provides a valuable resource for teaching material, too. The book is suitable for advanced undergraduate students in mathematics with a keen interest in (functional) analysis, for graduate students in analysis, ordinary differential equations or operator theory, and also as a reference for researchers. I am looking forward to reading the next books in this excellent series which creates a world, if such an unorthodox comparison be allowed, somehow similar in complexity to the (likewise lengthy) ``Game of Thrones'' saga. As previously announced, they shall be dealing with theory of partial differential equations, differential geometry, differentiable manifolds, and Lie groups.
Reviewer: Sorin-Mihai Grad (Wien)Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities.https://www.zbmath.org/1456.490112021-04-16T16:22:00+00:00"Hieu, Vu Trung"https://www.zbmath.org/authors/?q=ai:hieu.vu-trungSummary: In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities (VVIs). We show that if the weak Pareto solution set of a monotone VVI is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone VVIs are raised at the end of the paper.Norm inequalities for generalized Laplace transforms.https://www.zbmath.org/1456.440012021-04-16T16:22:00+00:00"Kuang, J. C."https://www.zbmath.org/authors/?q=ai:kuang.jianchao|kuang.jichang|kuang.jichang-cThe author obtains norm inequalities involving generalizations of Laplace, Stieltjes and Hankel transforms. The discrete versions of the norm inequalities are also investigated.
For the entire collection see [Zbl 1443.39001].
Reviewer: Osman Yürekli (Ithaca)Fixed point results for weak \(\varphi\)-contractions in cone metric spaces over Banach algebras and applications.https://www.zbmath.org/1456.540132021-04-16T16:22:00+00:00"Li, Biwen"https://www.zbmath.org/authors/?q=ai:li.biwen"Huang, Huaping"https://www.zbmath.org/authors/?q=ai:huang.huapingSummary: By using a nontrivial proof method, the purpose of this paper is to obtain some fixed point results for weak \(\varphi\)-contractions in cone metric spaces over Banach algebras. Several examples and applications to the existence and uniqueness of a solution to two classes of equations are also given.Extending representations of Banach algebras to their biduals.https://www.zbmath.org/1456.460422021-04-16T16:22:00+00:00"Gardella, Eusebio"https://www.zbmath.org/authors/?q=ai:gardella.eusebio"Thiel, Hannes"https://www.zbmath.org/authors/?q=ai:thiel.hannesLet \(\varphi\) be a representation of a Banach algebra \(A\) on a Banach space \(X\). The essential space \(X_\varphi\) of \(\varphi\) is the closed linear span of \{\(\varphi(a)x$,\, $a \in A$,\, $x \in X\)\}. Firstly, the authors give conditions so that there is an extension \(\widetilde{\varphi}\) of \(\varphi\) on the bidual (Banach) algebra of \(A\), under (the left or yet the right) Arens product; in that case, the essential spaces of \(\varphi\) and \(\widetilde{\varphi}\) agree. Based on this result, the authors deal with the complementarity of the essential spaces (with respect to representations of Banach algebras). A positive result is given when \(A\) has a bounded left approximate identity and every operator \(A \rightarrow X\) is weakly compact.
Further, the authors connect the existence of representations of certain Banach algebras to that of the respective (left) multiplier algebras. In particular, they consider the case when \(X\) belongs to a class of reflexive Banach spaces, closed under complementation (of its subspaces) and \(A\) is a certain representable Banach algebra with a contractive left approximate identity. Actually, \(A\) accepts a nondegenerate isometric representation on some space like \(X\). This implies that the left multiplier algebra of \(A\) has a unital,
isometric representation on a space \(X\), as before. This is also true for any \(C^\ast\)-algebra being isometrically representable on an \(L^p\)-space, \(p \in [1, \infty)\). Based, amongst others, on the latter result, the authors prove that, in the context of \(C^\ast\)-algebras \(A\), the (ring) commutativity characterizes such an \(A\) as isometrically represented on an \(L^p\)-space, \(p \in [1, \infty) \backslash \{2\}\).
Moreover, for a locally compact group \(G\), the completion of \(L^1(G)\) for nondegenerate representations
on \(L^p\)-spaces is called the universal group \(L^p\)-operator algebra of \(G\). Actually, the latter is universal for all contractive representations of \(L^1(G)\) on \(L^p\)-spaces.
Applications of the present paper are given by the authors in [Trans. Am. Math. Soc. 371, 2207--2236 (2019; Zbl 06999077)].
Reviewer: Marina Haralampidou (Athína)Discrete Cesaro operator between weighted Banach spaces on homogenous trees.https://www.zbmath.org/1456.460212021-04-16T16:22:00+00:00"Sharma, Ajay K."https://www.zbmath.org/authors/?q=ai:sharma.ajay-kumar|sharma.ayay-k"Kumar, Vivek"https://www.zbmath.org/authors/?q=ai:kumar.vivekThis paper studies the discrete Cesàro operator on weighted Banach spaces on homogenous
trees. Continuity and compactness are characterized. The discrete Cesàro operator acting on the Banach space of bounded functions on homogenous trees is continuous, but not compact. This operator acting on weighted Banach spaces of bounded functions on homogenous
trees may not be continuous for some weight functions. Moreover, examples of weight functions are presented such that the operator acting on the corresponding weighted Banach spaces of bounded functions on homogenous trees is compact.
Reviewer: José Bonet (Valencia)On existence theorems for generalized abstract measure integrodifferential equations.https://www.zbmath.org/1456.450022021-04-16T16:22:00+00:00"Dhage, Bapurao Chandrabhan"https://www.zbmath.org/authors/?q=ai:dhage.bapurao-chandrabhanSummary: In this paper, an existence and uniqueness results for a nonlinear abstract measure integrodifferential equation are proved via classical fixed point theorems of Schauder (see [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) and the author [Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 6, 9 p. (2002; Zbl 1029.47034)] under weaker Carathéodory condition. The existence for extremal solutions is also proved under certain Chandrabhan condition and using a hybrid fixed point theorem of the author [loc. cit.] in an ordered Banach space. Our existence results presented in this paper include the existence results of \textit{R. R. Sharma} [Proc. Am. Math. Soc. 32, 503--510 (1972; Zbl 0213.36201)], \textit{S. R. Joshi} [J. Math. Phys. Sci. 13, 497--506 (1979; Zbl 0435.34053)], \textit{G. R. Shendge} and \textit{S. R. Joshi} [Acta Math. Hung. 41, 53--59 (1983; Zbl 0536.34040)] and the author [J. Math. Phys. Sci. 20, 367--380 (1986; Zbl 0619.45005); with \textit{P. R. M. Reddy}, Jñānābha 49, No. 2, 82--93 (2019; Zbl 07273233)] on nonlinear abstract measure and abstract measure integrodifferential equations as special cases under weaker continuity condition.Spin-boson type models analyzed using symmetries.https://www.zbmath.org/1456.811782021-04-16T16:22:00+00:00"Dam, Thomas Norman"https://www.zbmath.org/authors/?q=ai:dam.thomas-norman"Møller, Jacob Schach"https://www.zbmath.org/authors/?q=ai:schach-moller.jacobThis paper is devoted to the investigation of a family of models for a qubit interacting with a bosonic field. The authors consider state space \(\mathbb{C}^2 \otimes\mathcal{F}_b(\mathcal{H})\), where Hilbert space \(\mathcal{H}\) is the state space of a single boson and \(\mathcal{F}_b(\mathcal{H})\) is the corresponding bosonic Fock space; the state space of the qubit is \(\mathbb{C}^2\). In the paper under review the following Hamiltonian is investigated
\[H_\eta(\alpha,f,\omega)= \eta\sigma_z\otimes 1 + 1\otimes d\Gamma(\omega)+\sum\limits_{i=1}^{2n} \alpha_i \left(\sigma_x\otimes \phi (f_i)\right)^i .\]
This operator is parameterized by \(\alpha\in\mathbb{C}^{2n}, f\in\mathcal{H}^{2n}, \eta\in\mathbb{C}\), \(\sigma_x, \sigma_y, \sigma_z\) denote the Pauli matrices, \(\omega\) is self-adjoint on \(\mathcal{H}\) and \(d\Gamma(\omega)\) is the second quantization of \(\omega\).
It is assumed that this Hamiltonian has a special symmetry, called \textit{spin-parity symmetry}. The spin-parity symmetry allows to find the domain of self-adjointness and decompose the Hamiltonian into two fiber operators each defined on Fock space. The authors prove the Hunziker-van Winter-Zhislin (HVZ) theorem for the fiber operators. The HVZ theorem for the fiber operators also gives an HVZ theorem for the full Hamiltonian. It is proved that if ground states exist for the full Hamiltonian, then the bottom of the spectrum is a nondegenerate
eigenvalue. Using this result, the authors single out a particular fiber, which has a ground state if and only if the full Hamiltonian has a ground state. Ground states for the other fiber operator must therefore correspond to excited states. A criterion for the existence of an excited state is also obtained.
Reviewer: Michael Perelmuter (Kyjiw)Common solution of Urysohn integral equations with the help of common fixed point results in complex valued metric spaces.https://www.zbmath.org/1456.540172021-04-16T16:22:00+00:00"Sintunavarat, Wutiphol"https://www.zbmath.org/authors/?q=ai:sintunavarat.wutiphol"Zada, Mian Bahadur"https://www.zbmath.org/authors/?q=ai:zada.mian-bahadur"Sarwar, Muhammad"https://www.zbmath.org/authors/?q=ai:sarwar.muhammadSummary: The aim of this manuscript is to discuss the existence and uniqueness of common solution for the following system of Urysohn integral equations:
\[
z(t)=\phi _{i}(t)+\int _{a}^{b}K_{i}(t,s,z(s)) ds,\eqno{(0.1)}
\]
where \(i=1,2,3,4\), \(a,b\in \mathbb {R}\) with \(a\leq b\), \(t\in [a,b]\), \(z, \phi _{i} \in C([a,b],\mathbb {R}^n)\) and \(K_{i}:[a,b]\times [a,b]\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is a given mapping for each \(i=1,2,3,4\). For this intention we establish common fixed point results for two pairs of weakly compatible mappings satisfying the contractive condition of rational type in the frame work of complex valued metric spaces.Study of the algebra of smooth integro-differential operators with applications.https://www.zbmath.org/1456.160212021-04-16T16:22:00+00:00"Haghany, A."https://www.zbmath.org/authors/?q=ai:haghany.ahmad"Kassaian, Adel"https://www.zbmath.org/authors/?q=ai:kassaian.adelReconstruction of convex bodies from moments.https://www.zbmath.org/1456.520052021-04-16T16:22:00+00:00"Kousholt, Astrid"https://www.zbmath.org/authors/?q=ai:kousholt.astrid"Schulte, Julia"https://www.zbmath.org/authors/?q=ai:schulte.juliaSummary: We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions.https://www.zbmath.org/1456.350832021-04-16T16:22:00+00:00"Kang, Hao"https://www.zbmath.org/authors/?q=ai:kang.hao"Ruan, Shigui"https://www.zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.A product expansion for Toeplitz operators on the Fock space.https://www.zbmath.org/1456.470072021-04-16T16:22:00+00:00"Hagger, Raffael"https://www.zbmath.org/authors/?q=ai:hagger.raffaelSummary: We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results, we require significantly fewer derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization [\textit{D. Borthwick}, Contemp. Math. 214, 23--37 (1998; Zbl 0903.58013)]. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.Existence of homoclinic orbits for a singular differential equation involving \(p\)-Laplacian.https://www.zbmath.org/1456.340452021-04-16T16:22:00+00:00"Yin, Honghui"https://www.zbmath.org/authors/?q=ai:yin.honghui"Du, Bo"https://www.zbmath.org/authors/?q=ai:du.bo"Yang, Qing"https://www.zbmath.org/authors/?q=ai:yang.qing"Duan, Feng"https://www.zbmath.org/authors/?q=ai:duan.fengIn the present manuscript, the authors are concerned with the existence of {homoclinic} solutions for the following singular ODE
\[
\Big(\Phi_p\big(x'(t)\big)\Big)'+f\big(x'(t)\big) + g\big(x(t)\big) + \frac{h(t)}{1-x(t)} = e(t), \tag{1}
\]
where $\Phi_p(s) = |s|^{p-2}s$ (for some $p > 1$), $f,g,h,e\in C(\mathbb{R};\mathbb{R})$ and, moreover, $h$ is a strictly positive $T$-periodic function.
As usual, a \textit{homoclinic solution} of (1) is a solution $x\in C(\mathbb{R};\mathbb{R})$ satisfying
\[
\text{$x(t)\to\infty$ as $|t|\to\infty$}.
\]
Due to their relevance in several contexts, homoclinic solutions for general differential systems have been studied by many authors and with different techniques (variational methods, critical-point theory, method of lower/upper solutions and fixed-point theorems, etc.); however, since equation (1) is strongly nonlinear, these traditional techniques are no-longer applicable.
Using a new continuation theorem due to Manásevich and Mawhin, the authors obtain the following theorem, which is the main result of the paper.
Theorem 1.
Assume that the following assumptions are satisfied:
\begin{itemize}
\item[{(H.1)}] $f:\mathbb{R}\to\mathbb{R}$ is continuous, bounded and non-negative;
\item[{(H.2)}] $g:\mathbb{R}\to\mathbb{R}$ is strictly monotone increasing and there are positive constants $\sigma$ and $n$ such that
\[
xg(x)\geq \sigma|x|^{n+1}\quad\text{for all $x\in\mathbb{R}$};
\]
\item[{(H.3)}] $\rho_1 := \sup_{t\in\mathbb{R}}|e(t)| < \infty$ and
\[
\rho_2 := \int_{\mathbb{R}}|e(t)|^{1+1/n}\,\mathrm{d} t < \infty.
\]
\end{itemize}
Then, if $\rho_1 > f(0)$ and $h_l/\rho_1 - f(0) < 1$ (with $h_l := \min_{t\in\mathbb{R}}h(t)$), there exists at least one positive homoclinic solution $\omega_0$, further satisfying
\[
|\omega_0'(t)|\to 0\quad\text{as $|t|\to\infty$}.
\]
Thought it is based on the continuation theorem by Manásevich and Mawhin, the proof of Theorem 1 is sophisticated and it requires some preliminary lemmas of independent interest. On the other hand, a couple of examples at the end of the paper show the wide range of applicability of this result.
Reviewer: Stefano Biagi (Milano)On the attainability of the best constant in fractional Hardy-Sobolev inequalities involving the spectral Dirichlet Laplacian.https://www.zbmath.org/1456.460352021-04-16T16:22:00+00:00"Ustinov, N. S."https://www.zbmath.org/authors/?q=ai:ustinov.n-sSummary: We prove the attainability of the best constant in the fractional Hardy-Sobolev inequality with a boundary singularity for the spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin. A similar result has been proved earlier for the conventional Hardy-Sobolev inequality [\textit{A. V. Dem'yanov} and \textit{A. I. Nazarov}, Zap. Nauchn. Semin. POMI 336, 25--45 (2006; Zbl 1136.35088); translation in J. Math. Sci., New York 143, No. 2, 2857--2868 (2007)].Multilinear operators between asymmetric normed spaces.https://www.zbmath.org/1456.460412021-04-16T16:22:00+00:00"Latreche, Faiz"https://www.zbmath.org/authors/?q=ai:latreche.faiz"Dahia, Elhadj"https://www.zbmath.org/authors/?q=ai:dahia.elhadjThe authors prove some fundamental results for multilinear operators between asymmetric normed spaces (see [\textit{S. Cobzaş}, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]). Among other results, they give criteria for the continuity of multilinear operators, Banach-Steinhaus type theorems, and a closed graph theorem. These extend the corresponding results for multilinear operators between normed spaces as well as those for linear operators between asymmetric normed spaces, as, for instance, those proved in [\textit{M. D. Mabula} and \textit{S. Cobzaş}, Topology Appl. 184, 1--15 (2015; Zbl 1322.46004)].
Reviewer: Stefan Cobzaş (Cluj-Napoca)Convergence to approximate solutions and perturbation resilience of iterative algorithms.https://www.zbmath.org/1456.470302021-04-16T16:22:00+00:00"Reich, Simeon"https://www.zbmath.org/authors/?q=ai:reich.simeon"Zaslavski, Alexander J."https://www.zbmath.org/authors/?q=ai:zaslavski.alexander-jAuthors' abstract: We first consider nonexpansive self-mappings of a metric space and study the asymptotic behavior of their inexact orbits. We then apply our results to the analysis of iterative methods for finding approximate fixed points of nonexpansive mappings and approximate zeros of monotone operators.
Reviewer: Boro Döring (Düsseldorf)Characterizations of \({*}\) and \({*}\)-left derivable mappings on some algebras.https://www.zbmath.org/1456.460562021-04-16T16:22:00+00:00"An, Guangyu"https://www.zbmath.org/authors/?q=ai:an.guangyu"He, Jun"https://www.zbmath.org/authors/?q=ai:he.jun"Li, Jiankui"https://www.zbmath.org/authors/?q=ai:li.jiankuiSummary: A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*}$-$\mathcal{A} \)-bimodule \(\mathcal{M}\) is a \({*} \)-derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+\delta (A)B=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every (continuous) \({*} \)-derivable mapping at \(G\) from a (unital \(C^*\)-algebra) factor von Neumann algebra into its Banach \({*}\)-bimodule is a \({*}\)-derivation if and only if \(G\) is a left separating point. A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*} \)-left \(\mathcal{A} \)-module \(\mathcal{M}\) is a \({*} \)-left derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+B\delta (A)=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every continuous \({*} \)-left derivable mapping at a left separating point from a unital \(C^*\)-algebra or von Neumann algebra into its Banach \({*}\)-left \(\mathcal{A} \)-module is identical with zero under certain conditions.Periodic solutions for the Lorentz force equation with singular potentials.https://www.zbmath.org/1456.340412021-04-16T16:22:00+00:00"Garzón, Manuel"https://www.zbmath.org/authors/?q=ai:garzon.manuel"Torres, Pedro J."https://www.zbmath.org/authors/?q=ai:torres.pedro-joseThe authors consider the Lorentz force equation
\[
(\phi(q'))'=E(t,q)+q'\times B(t,q)\,,
\]
where \(E\) and \(B\) are the electric and magnetic field, respectively, and
\[
\phi(v)=\frac{v}{\sqrt{1-|v|^2}}\,.
\]
They take \(E(t,q)=-\nabla V(q)+h(t)\), with \(h\in L^1([0,T],\mathbb{R}^3)\), and they assume
(H1) \(V\in C^1(\mathbb{R}^3\setminus\{0\},R)\) and
\[\lim_{|q|\to\infty}\nabla V(q)=0;\]
(H2) \(q\cdot \nabla V(q)<0\) for every \(q\in \mathbb{R}^3\) and there exist \(c_0,\varepsilon_0>0\) and \(\gamma\ge1\) such that \(q\cdot\nabla V(q)\le-c_0|q|^{-\gamma}\) when \(|q|<\varepsilon_0\);
(H3) \(B\in C([0,T]\times \mathbb{R}^3\setminus\{0\},\mathbb{R}^3)\) and there exists \(C_B>0\) such that \[\limsup_{|q|\to\infty}|B(t,q)|<C_B<\frac{1}{T}\Big|\int_0^Th(t)\,dt\Big|;\]
(H4) there exist \(c_1,\varepsilon_1>0\) and \(\beta\in(0,\gamma)\) such that \(|B(t,q)|\le c_1|q|^{-\beta-1}\) for all \(t\in[0,T]\) when \(|q|<\varepsilon_1\).
Under the above assumptions, by the use of topological degree methods they prove that the Lorenz force equation has at least one solution \(q:[0,T]\to \mathbb{R}^3\) such that \(|q'(t)|<1\) for every \(t\in[0,T]\) and \(q(0)-q(T)=0=q'(0)-q'(T)\).
Reviewer: Alessandro Fonda (Trieste)Multiple periodic solutions for a Duffing type equation with one-sided sublinear nonlinearity: beyond the Poincaré-Birkhoff twist theorem.https://www.zbmath.org/1456.340402021-04-16T16:22:00+00:00"Dondè, Tobia"https://www.zbmath.org/authors/?q=ai:donde.tobia"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioSummary: We prove the existence of multiple periodic solutions for a planar Hamiltonian system generated from the second order scalar ODE of Duffing type \(x'' + q(t)g(x) = 0\) with \(g\) satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. We focus our analysis to the case of a stepwise weight function, in order to highlight the underlying geometrical structure.
For the entire collection see [Zbl 1445.34003].On the asymptotic dynamics of 2-D magnetic quantum systems.https://www.zbmath.org/1456.811762021-04-16T16:22:00+00:00"Cárdenas, Esteban"https://www.zbmath.org/authors/?q=ai:cardenas.esteban"Hundertmark, Dirk"https://www.zbmath.org/authors/?q=ai:hundertmark.dirk"Stockmeyer, Edgardo"https://www.zbmath.org/authors/?q=ai:stockmeyer.edgardo"Vugalter, Semjon"https://www.zbmath.org/authors/?q=ai:vugalter.semjon-aSummary: In this work, we provide results on the long-time localization in space (dynamical localization) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form \(H = H_0 + W\), where \(H_0\) is rotationally symmetric and has dense point spectrum and \(W\) is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.Chiral Floquet systems and quantum walks at half-period.https://www.zbmath.org/1456.811962021-04-16T16:22:00+00:00"Cedzich, C."https://www.zbmath.org/authors/?q=ai:cedzich.christopher"Geib, T."https://www.zbmath.org/authors/?q=ai:geib.t"Werner, A. H."https://www.zbmath.org/authors/?q=ai:werner.albert-h"Werner, R. F."https://www.zbmath.org/authors/?q=ai:werner.reinhard-fSummary: We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous, or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices, it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half-period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory, we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at \(+1\) and \(-1\) which is not possible for a single timeframe.Matrix Richard inequality via the geometric mean.https://www.zbmath.org/1456.150182021-04-16T16:22:00+00:00"Fujimoto, Masayuki"https://www.zbmath.org/authors/?q=ai:fujimoto.masayuki"Seo, Yuki"https://www.zbmath.org/authors/?q=ai:seo.yukiSummary: In this paper, we show the matrix version of Richard inequality by virture of Cauchy-Schwartz type inequalities via the matrix geometric mean. As an application, we show a matrix Buzano inequality.Analysis and stochastic processes on metric measure spaces.https://www.zbmath.org/1456.580192021-04-16T16:22:00+00:00"Grigor'yan, Alexander"https://www.zbmath.org/authors/?q=ai:grigoryan.alexanderThe purpose of the author is to survey some known results of the Laplacian operator on a geodesically complete and non-compact Riemannian manifold. Precisely, the overview contains, e.g., Semi-linear elliptic inequalities, Negative eigenvalues of Schrödinger, Estimates of the Green function, Heat kernels on connected sums, of Schrödinger operator, and of operators with singular drift, and so on. Likewise, the author deals with sections on Analysis on metric measure spaces and on Homology theory on graphs.
For the entire collection see [Zbl 1416.60012].
Reviewer: Mohammed El Aïdi (Bogotá)A mathematical analysis of the GW\(^0\) method for computing electronic excited energies of molecules.https://www.zbmath.org/1456.811852021-04-16T16:22:00+00:00"Cancés, Eric"https://www.zbmath.org/authors/?q=ai:cances.eric"Gontier, David"https://www.zbmath.org/authors/?q=ai:gontier.david"Stoltz, Gabriel"https://www.zbmath.org/authors/?q=ai:stoltz.gabrielSpectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition.https://www.zbmath.org/1456.470152021-04-16T16:22:00+00:00"Li, Kun"https://www.zbmath.org/authors/?q=ai:li.kun.2"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiong"Hao, Xiaoling"https://www.zbmath.org/authors/?q=ai:hao.xiaoling"Bao, Qinglan"https://www.zbmath.org/authors/?q=ai:bao.qinglanSummary: In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator \(L_h\) by using operator theoretic formulation. The self-adjoint dilation \(\mathcal{T}_h\) of \(L_h\) in the space \(\mathcal{H}\) is constructed, furthermore, the incoming and outgoing representations of \(\mathcal{T}_h\) and functional model are also constructed, hence in light of Lax-Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, a completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved.Hilbert matrix on spaces of Bergman-type.https://www.zbmath.org/1456.470112021-04-16T16:22:00+00:00"Jevtić, Miroljub"https://www.zbmath.org/authors/?q=ai:jevtic.miroljub"Karapetrović, Boban"https://www.zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: It is well known (see [\textit{M. Jevtić} and \textit{B. Karapetrović}, ``Libera operator on mixed norm spaces \(H_\nu^{p, q, \alpha}\) when \(0 <p < 1\)'', Filomat 31, No. 14, 4641--4650 (2017; \url{doi:10.2298/FIL1714641J}); \textit{M. Pavlović}, ``Definition and properties of the Libera operator on mixed norm spaces'', The Scientific World Journal 2014, Article ID 590656, 15 p. (2014; \url{doi:10.1155/2014/590656})]) that the Libera operator \(\mathcal{L}\) is bounded on the Besov space \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} : = \nu - \alpha - \frac{1}{p} + 1\). We prove unexpected results: the Hilbert matrix operator \(H\), as well as the modified Hilbert operator \(\tilde{H}\), is bounded on \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} < 1\). In particular, \(H\), as well as \(\tilde{H}\), is bounded on the Bergman space \(A^{p, \alpha}\) if and only if \(1 < \alpha + 2 < p\) and is bounded on the Dirichlet space \(\mathcal{D}_\alpha^p = A_1^{p, \alpha}\) if and only if \(\max \{- 1, p - 2 \} < \alpha < 2 p - 2\). Our results are substantial improvement of [\textit{B. Łanucha} et al., Ann. Acad. Sci. Fenn., Math. 37, No. 1, 161--174 (2012; Zbl 1258.47047), Theorem 3.1] and of [\textit{P. Galanopoulos} et al., Ann. Acad. Sci. Fenn., Math. 39, No. 1, 231--258 (2014; Zbl 1297.47030), Theorem 5].The dimensional Brunn-Minkowski inequality in Gauss space.https://www.zbmath.org/1456.520112021-04-16T16:22:00+00:00"Eskenazis, Alexandros"https://www.zbmath.org/authors/?q=ai:eskenazis.alexandros"Moschidis, Georgios"https://www.zbmath.org/authors/?q=ai:moschidis.georgiosThe authors prove the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure, thus settling a problem raised in [\textit{R. J. Gardner} and \textit{A. Zvavitch}, Trans. Am. Math. Soc. 362, No. 10, 5333--5353 (2010; Zbl 1205.52002)]. They also settle the case when equality holds.
Reviewer: George Stoica (Saint John)Existence of solutions of fuzzy fractional panto-graph equations.https://www.zbmath.org/1456.340772021-04-16T16:22:00+00:00"Agilan, K."https://www.zbmath.org/authors/?q=ai:agilan.k"Parthiban, V."https://www.zbmath.org/authors/?q=ai:parthiban.vijayaAuthors consider nonlinear fuzzy fractional panto-graph equation with the Caputo gH-derivative
\[(_{gH}D^q_{0+})u(x)=f(x,u(x),u(\lambda x)), \quad u(0)=u_0,\]
where \(0 < q \leq 1\) is a real number and the operator \( (_{gH}D^q_{0+})\) indicate the Caputo fractional generalized derivative of order \(q,\) \(f: J\times R_f\times R_f\to R_f\) is the continuous function. Conditions for the existence of a solution are obtained. Finally, the authors give an example to support the results.
Reviewer: Tatyana Komleva (Odessa)Complex function theory, operator theory, Schur analysis and systems theory. A volume in honor of V. E. Katsnelson.https://www.zbmath.org/1456.001062021-04-16T16:22:00+00:00"Alpay, Daniel (ed.)"https://www.zbmath.org/authors/?q=ai:alpay.daniel"Fritzsche, Bernd (ed.)"https://www.zbmath.org/authors/?q=ai:fritzsche.bernd"Kirstein, Bernd (ed.)"https://www.zbmath.org/authors/?q=ai:kirstein.berndPublisher's description: This book is dedicated to Victor Emmanuilovich Katsnelson on the occasion of his 75th birthday and celebrates his broad mathematical interests and contributions. Victor Emmanuilovich's mathematical career has been based mainly at the Kharkov University and the Weizmann Institute. However, it also included a one-year guest professorship at Leipzig University in 1991, which led to him establishing close research contacts with the Schur analysis group in Leipzig, a collaboration that still continues today.
Reflecting these three periods in Victor Emmanuilovich's career, present and former colleagues have contributed to this book with research inspired by him and presentations on their joint work. Contributions include papers in function theory (Favorov-Golinskii, Friedland-Goldman-Yomdin, Kheifets-Yuditskii), Schur analysis, moment problems and related topics (Boiko-Dubovoy, Dyukarev, Fritzsche-Kirstein-Mädler), extension of linear operators and linear relations (Dijksma-Langer, Hassi-de Snoo, Hassi -Wietsma) and non-commutative analysis (Ball-Bolotnikov, Cho-Jorgensen).
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Alpay, Daniel (ed.); Fritzsche, Bernd (ed.); Kirstein, Bernd (ed.)}, Editorial introduction, 1-8 [Zbl 07310471]
\textit{Dym, Harry}, Victor comes to Rehovot, 11-13 [Zbl 07310472]
\textit{Dyukarev, Yu. M.}, My teacher Viktor Emmanuilovich Katsnelson, 15-17 [Zbl 07310473]
\textit{Feldman, G. M.}, Some impressions of Viktor Emmanuilovich Katsnelson, 19-23 [Zbl 07310474]
\textit{Kirstein, Bernd}, The good fortune of maintaining a long-lasting close friendship and scientific collaboration with V. E. Katsnelson, 25-60 [Zbl 07310475]
\textit{Sodin, Mikhail}, A piece of Victor Katsnelson's mathematical biography, 61-75 [Zbl 07310476]
\textit{Ball, Joseph A.; Bolotnikov, Vladimir}, Interpolation by contractive multipliers between Fock spaces, 79-138 [Zbl 07310478]
\textit{Boiko, S. S.; Dubovoy, V. K.}, Regular extensions and defect functions of contractive measurable operator-valued functions, 139-228 [Zbl 07310479]
\textit{Cho, Ilwoo; Jorgensen, Palle}, Free-homomorphic relations induced by certain free semicircular families, 229-285 [Zbl 07310480]
\textit{Dijksma, Aad; Langer, Heinz}, Self-adjoint extensions of a symmetric linear relation with finite defect: compressions and Straus subspaces, 287-325 [Zbl 07310481]
\textit{Dyukarev, Yu. M.}, On conditions of complete indeterminacy for the matricial Hamburger moment problem, 327-353 [Zbl 07310482]
\textit{Favorov, S.; Golinskii, L.}, On a Blaschke-type condition for subharmonic functions with two sets of singularities on the boundary, 355-375 [Zbl 07310483]
\textit{Friedland, Omer; Goldman, Gil; Yomdin, Yosef}, Exponential Taylor domination, 377-386 [Zbl 07310484]
\textit{Fritzsche, Bernd; Kirstein, Bernd; Mädler, Conrad}, A closer look at the solution set of the truncated matricial moment problem \(\mathsf{M}[\alpha,\infty);(s_{j})_{j=0}^{m},{\preccurlyeq} ]\), 387-492 [Zbl 07310485]
\textit{Hassi, Seppo; De Snoo, H. S. V.}, A class of sectorial relations and the associated closed forms, 493-514 [Zbl 07310486]
\textit{Hassi, Seppo; Wietsma, Hendrik Luit}, Spectral decompositions of selfadjoint relations in Pontryagin spaces and factorizations of generalized Nevanlinna functions, 515-534 [Zbl 07310487]
\textit{Kheifets, A.; Yuditskii, P.}, Martin functions of Fuchsian groups and character automorphic subspaces of the Hardy space in the upper half plane, 535-581 [Zbl 07310488]Spectral analysis of polygonal cavities containing a negative-index material.https://www.zbmath.org/1456.351442021-04-16T16:22:00+00:00"Hazard, Christophe"https://www.zbmath.org/authors/?q=ai:hazard.christophe"Paolantoni, Sandrine"https://www.zbmath.org/authors/?q=ai:paolantoni.sandrineSummary: The purpose of this paper is to investigate the spectral effects of an interface between vacuum and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM: the non dissipative Drude model, for which negativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the complete Maxwell's equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled with a portion of Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and we present a complete description of the spectrum. We show in particular that the interface between the NIM and vacuum is responsible for various resonance phenomena related to various components of an \textit{essential spectrum}.Pseudospectrum of an element of a Banach algebra.https://www.zbmath.org/1456.470022021-04-16T16:22:00+00:00"Krishnan, Arundhathi"https://www.zbmath.org/authors/?q=ai:krishnan.arundhathi"Kulkarni, S. H."https://www.zbmath.org/authors/?q=ai:kulkarni.s-hSummary: The \(\varepsilon\)-pseudospectrum \(\Lambda_\varepsilon(a)\) of an element \(a\) of an arbitrary Banach algebra \(A\) is studied. Its relationships with the spectrum and numerical range of \(a\) are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of \(a\). Suppose for some \(\epsilon > 0\) and \(a,b\in A\), \(\Lambda_\epsilon(ax) = \Lambda_\epsilon(bx)\) for all \(x \in A\). It is shown that \(a=b\) if:
\begin{itemize}
\item[(i)] \(a\) is invertible.
\item[(ii)] \(a\) is Hermitian idempotent.
\item[(iii)] \(a\) is the product of a Hermitian idempotent and an invertible element.
\item[(iv)] \(A\) is semisimple and \(a\) is the product of an idempotent and an invertible element.
\item[(v)] \(A=B(X)\) for a Banach space \(X\).
\item[(vi)] \(A\) is a \(C^*\)-algebra.
\item[(vii)]\(A\) is a commutative semisimple Banach algebra.
\end{itemize}On a final value problem for a nonlinear fractional pseudo-parabolic equation.https://www.zbmath.org/1456.351142021-04-16T16:22:00+00:00"Au, Vo Van"https://www.zbmath.org/authors/?q=ai:au.vo-van"Jafari, Hossein"https://www.zbmath.org/authors/?q=ai:jafari.hossein"Hammouch, Zakia"https://www.zbmath.org/authors/?q=ai:hammouch.zakia"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: In this paper, we investigate a final boundary value problem for a class of fractional with parameter \(\beta\) pseudo-parabolic partial differential equations with nonlinear reaction term. For \(0<\beta <1\), the solution is regularity-loss, we establish the well-posedness of solutions. In the case that \(\beta >1\), it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.An operator-valued \(T(1)\) theorem for symmetric singular integrals in UMD spaces.https://www.zbmath.org/1456.420172021-04-16T16:22:00+00:00"Hytönen, Tuomas"https://www.zbmath.org/authors/?q=ai:hytonen.tuomas-pSummary: The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``\(T(1) \in \text{BMO}\)'' assumptions that are not easily checkable. Recently, \textit{G. Hong} et al. [J. Funct. Anal. 278, No. 7, Article ID 108420, 27 p. (2020; Zbl 07155094)] observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical \(T(1)\) criterion still guarantees their \(L^2\)-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by \(R\)-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative \(L^p\) spaces for all \(1 < p < \infty\), without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of G. Hong et al. [loc. cit.] for \(p \neq 2\).Hyers-Ulam stability of a nonautonomous semilinear equation with fractional diffusion.https://www.zbmath.org/1456.352242021-04-16T16:22:00+00:00"Villa-Morales, José"https://www.zbmath.org/authors/?q=ai:villa-morales.joseSummary: In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities.https://www.zbmath.org/1456.580242021-04-16T16:22:00+00:00"Kalvin, Victor"https://www.zbmath.org/authors/?q=ai:kalvin.victorSummary: We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area \(4 \pi\) the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions.https://www.zbmath.org/1456.340042021-04-16T16:22:00+00:00"Derbazi, Choukri"https://www.zbmath.org/authors/?q=ai:derbazi.choukri"Hammouche, Hadda"https://www.zbmath.org/authors/?q=ai:hammouche.haddaSummary: In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O'Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results.Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems.https://www.zbmath.org/1456.352252021-04-16T16:22:00+00:00"Wang, Fuliang"https://www.zbmath.org/authors/?q=ai:wang.fuliang"Die, Hu"https://www.zbmath.org/authors/?q=ai:die.hu"Xiang, Mingqi"https://www.zbmath.org/authors/?q=ai:xiang.mingqiSummary: In this paper, we consider the existence and multiplicity of solutions for the following fractional Laplacian system with logarithmic nonlinearity
\[\begin{cases}
(-\Delta)^su=\lambda h_1(x)u\ln |u|+\frac{p}{p+q}b(x)|v|^q|u|^{p-2}u\quad & x\in \Omega, \\
(-\Delta)^t v=\mu h_2(x)v\ln |v|+\frac{q}{p+q} b(x) |u|^p|v|^{q-2} v \quad & x\in \Omega,\\
u=v=0 \quad & x\in \mathbb{R}^N \setminus \Omega, \end{cases}
\]
where \(s\), \(t\in (0,1)\), \(N>\max \{2s,2t\}\), \(\lambda,\), \(\mu >0\), \(2<p+q<\min \{\frac{2N}{N-2s}\), \(\frac{2N}{N-2t}\}\), \(\Omega \subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary, \(h_1\), \(h_2\), \(b\in C (\overline{\Omega})\) and \((-\Delta)^s\) is the fractional Laplacian. When \(h_1\), \(h_2\), \(b\) are positive functions, the existence of ground state solutions for the problem is obtained. When \(h_1\), \(h_2\) are sign-changing functions and \(b\) is a positive function, two nontrivial and nonnegative solutions are obtained. Our results are new even in the case of a single equation.Corrigendum to: ``Completely bounded norms of right module maps''.https://www.zbmath.org/1456.460472021-04-16T16:22:00+00:00"Levene, Rupert H."https://www.zbmath.org/authors/?q=ai:levene.rupert-h"Timoney, Richard M."https://www.zbmath.org/authors/?q=ai:timoney.richard-mThis correction refers to [the authors, ibid. 436, No. 5, 1406--1424 (2012; Zbl 1244.46026)].The FPP for left reversible semigroups in \(\ell_1\) endowed with different locally convex topologies.https://www.zbmath.org/1456.460182021-04-16T16:22:00+00:00"Japón, Maria A."https://www.zbmath.org/authors/?q=ai:japon.maria-aSummary: In this article we will consider locally convex topologies \(\tau\) on \(\ell_1\) which are coarser than the weak topology on the unit ball and such that the unit vector basic sequence \((e_n)\) is \(\tau\)-convergent. We characterize these topologies depending on the \(\tau\)-fixed point property for left reversible semigroups on \((\ell_1,\|\cdot\|_1)\). We will apply our results to the case of different weak* topologies on \(\ell_1\).Strong convergence theorems of iterative algorithm for nonconvex variational inequalities.https://www.zbmath.org/1456.470262021-04-16T16:22:00+00:00"Inchan, Issara"https://www.zbmath.org/authors/?q=ai:inchan.issaraSummary: In this work, we suggest and analyze an iterative scheme for solving a system of nonconvex variational inequalities by using projection technique. We prove strong convergence of iterative scheme to a solution of asystem of nonconvex variational inequalities requires the modified mapping \(T\) be Lipschitz continuous but not strongly monotone. Our result can be viewed as improvement of the result of \textit{N. Petrot} [Abstr. Appl. Anal. 2010, Article ID 472760, 9 p. (2010; Zbl 1206.49012)].Large orbits of operators and operator semigroups.https://www.zbmath.org/1456.470122021-04-16T16:22:00+00:00"Müller, Vladimir"https://www.zbmath.org/authors/?q=ai:muller.vladimirSummary: The paper represents a survey of results concerning `large' orbits of operators and strongly continuous operator semigroups.
For the entire collection see [Zbl 1300.47008].Some results on Bregman totally asymptotically strict quasi-pseudo-contractions.https://www.zbmath.org/1456.470332021-04-16T16:22:00+00:00"Liu, Qingmin"https://www.zbmath.org/authors/?q=ai:liu.qingmin"Wang, Zi-Ming"https://www.zbmath.org/authors/?q=ai:wang.zi-ming|wang.ziming"Wei, Airong"https://www.zbmath.org/authors/?q=ai:wei.airongSummary: The paper addresses both design and convergence analysis of fixed point iterative algorithms for a new nonlinear operator called Bregman totally asymptotically strict quasi-pseudo-contraction. Strong convergence theorems are established in real reflexive Banach spaces.Strong convergence of a generalized-projection method in Banach spaces.https://www.zbmath.org/1456.470252021-04-16T16:22:00+00:00"Hao, Yan"https://www.zbmath.org/authors/?q=ai:hao.yanSummary: A generalized-projection method is introduced for solving a generalized equilibrium problem, due to \textit{S. Takahashi} and \textit{W. Takahashi} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 3, 1025--1033 (2008; Zbl 1142.47350)], and a common fixed point problem of a family of generalized asymptotically quasi-\(\phi\)-nonexpansive mappings. It is proved that the generalized-projection method is strongly convergent in a Banach space.Iterative algorithms with Armijo-like search for solving split equality problems.https://www.zbmath.org/1456.470242021-04-16T16:22:00+00:00"Fu, Yuanmin"https://www.zbmath.org/authors/?q=ai:fu.yuanmin"Zhu, Li-Jun"https://www.zbmath.org/authors/?q=ai:zhu.lijun"He, Long"https://www.zbmath.org/authors/?q=ai:he.longSummary: In this paper, we propose two algorithms with Armijo-like search for solving split equality problems, furthermore, the second algorithm is constructed by forward-backward splitting method. Convergence results of the two algorithms are obtained.Zero and fixed points in Banach spaces.https://www.zbmath.org/1456.470232021-04-16T16:22:00+00:00"Cho, Sun Young"https://www.zbmath.org/authors/?q=ai:cho.sun-youngSummary: In this paper, we investigate a forward-backward splitting algorithm for fixed points of strict pseudocontractions and zero points of the sum of an \(m\)-accretive operator and an inverse-strongly accretive operator. Weak convergence theorems of common solutions are established in the framework of \(q\)-uniformly smooth Banach spaces.A hybrid steepest descent method for a split feasibility problem in Hilbert spaces.https://www.zbmath.org/1456.470222021-04-16T16:22:00+00:00"Cheng, Peng"https://www.zbmath.org/authors/?q=ai:cheng.peng.2|cheng.peng|cheng.peng.1Summary: The purpose of this paper is to introduce and investigate a hybrid steepest descent method for solving a split feasibility problem involving two bifunctions. It is proved that the solution of the split feasibility problem is also a unique solution to some strongly monotone variational inequality.Generalized mixed equilibrium problems and fixed point problems.https://www.zbmath.org/1456.470172021-04-16T16:22:00+00:00"Cheng, Peng"https://www.zbmath.org/authors/?q=ai:cheng.peng.2Summary: The purpose of this paper is to introduce and study fixed points of relatively generalized asymptotically nonexpansive mappings and solutions of generalized mixed equilibrium problems. We establish a weak convergence theorem to common solutions in a uniformly convex and uniformly smooth Banach space.Iterative approximation solvability of Bregman operator equations in reflexive Banach spaces.https://www.zbmath.org/1456.470212021-04-16T16:22:00+00:00"Al-Mazrooei, A. E."https://www.zbmath.org/authors/?q=ai:al-mazrooei.abdullah-eqal"Latif, A."https://www.zbmath.org/authors/?q=ai:latif.abdul"Qin, X."https://www.zbmath.org/authors/?q=ai:qin.xiaolongSummary: In this paper, we introduce a new nonlinear operator, the Bregman asymptotically strict quasi-pseudocontraction, and a simple Bregman shrinking projection algorithm. Strong convergence of the projection algorithm is obtained in reflexive Banach spaces. As an application, a system of generalized mixed equilibrium problems is considered.Spectral properties of Toeplitz operators on the unit ball and on the unit sphere.https://www.zbmath.org/1456.470062021-04-16T16:22:00+00:00"Akkar, Zineb"https://www.zbmath.org/authors/?q=ai:akkar.zineb"Albrecht, Ernst"https://www.zbmath.org/authors/?q=ai:albrecht.ernstSummary: In this article, we consider Toeplitz operators on the Hardy and weighted Bergman Hilbert spaces of the unit sphere, respectively on the unit ball in $\mathbb{C}^N$. Various aspects of the interplay between local and global properties of the symbols and local and global spectral properties of the corresponding Toeplitz operators are investigated. A local version of the spectral inclusion theorem of \textit{A. M. Davie} and \textit{N. P. Jewell} [J. Funct. Anal. 26, 356--368 (1977; Zbl 0374.47011)] is proved. Using some recent results of \textit{R. Quiroga-Barranco} and \textit{N. Vasilevski} [Integral Equations Oper. Theory 59, No. 3, 379--419 (2007; Zbl 1144.47024); ibid. 60, No. 1, 89--132 (2008; Zbl 1144.47025)], we describe some commutative $C^*$-subalgebras of the Toeplitz algebra for $N \geq2$. The method of \textit{G. McDonald} [Ill. J. Math. 23, 286--293 (1979; Zbl 0438.47031)] to compute the essential spectrum of Toeplitz operators with certain piecewise continuous symbols is extended to a larger class of symbols including examples where the surface measure of set of discontinuity points has strictly positive measure.
For the entire collection see [Zbl 1300.47008].A class of differential inverse variational inequalities in finite dimensional spaces.https://www.zbmath.org/1456.490102021-04-16T16:22:00+00:00"Feng, Jun"https://www.zbmath.org/authors/?q=ai:feng.jun"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11"Chen, Hui"https://www.zbmath.org/authors/?q=ai:chen.hui"Chen, Yuanchun"https://www.zbmath.org/authors/?q=ai:chen.yuanchunSummary: In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Carathéodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.Toeplitz and asymptotic Toeplitz operators on \(H^2(\mathbb{D}^n)\).https://www.zbmath.org/1456.470082021-04-16T16:22:00+00:00"Maji, Amit"https://www.zbmath.org/authors/?q=ai:maji.amit"Sarkar, Jaydeb"https://www.zbmath.org/authors/?q=ai:sarkar.jaydeb"Sarkar, Srijan"https://www.zbmath.org/authors/?q=ai:sarkar.srijanSummary: We initiate a study of Toeplitz operators and asymptotic Toeplitz operators on the Hardy space \(H^2(\mathbb{D}^n)\) (over the unit polydisc \(\mathbb{D}^n\) in \(\mathbb{C}^n\)). Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let \(T_{z_i}\) denote the multiplication operator on \(H^2(\mathbb{D}^n)\) by the \(i\)-th coordinate function \(z_i\), \(i = 1, \dots, n\), and let \(T\) be a bounded linear operator on \(H^2(\mathbb{D}^n)\). Then the following hold: \begin{itemize}\item[(i)] \(T\) is a Toeplitz operator (that is, \(T = P_{H^2(\mathbb{D}^n)} M_\varphi |_{H^2(\mathbb{D}^n)}\), where \(M_\varphi\) is the Laurent operator on \(L^2(\mathbb{T}^n)\) for some \(\varphi \in L^\infty(\mathbb{T}^n)\)) if and only if \(T_{z_i}^\ast T T_{z_i} = T\) for all \(i = 1, \dots, n\).\item[(ii)] \(T\) is an asymptotic Toeplitz operator if and only if \(T = \text{Toeplitz} + \text{compact}\).
\end{itemize}
The case \(n = 1\) gives the well-known results of \textit{A. Brown} and \textit{P. R. Halmos} [J. Reine Angew. Math. 213, 89--102 (1963; Zbl 0116.32501)] and \textit{A. Feintuch} [Oper. Theory, Adv. Appl. 41, 241--254 (1989; Zbl 0676.47014)], respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.On strong convergence of a Halpern-Mann's type iteration with perturbations for common fixed point and generalized equilibrium problems.https://www.zbmath.org/1456.470322021-04-16T16:22:00+00:00"Wattanataweekul, Manakorn"https://www.zbmath.org/authors/?q=ai:wattanataweekul.manakornSummary: We establish strong convergence of a sequence generated by a Halpern-Mann's type iteration with perturbation for approximating a common element of the set of fixed points of a countable family of quasi-nonexpansive mappings and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Within an appropriate setting, some results for solving the minimum-norm problems are also included. Finally, we consider the modified viscosity method of a countable family of nonexpansive mappings. The results presented in this paper extend and improve previously known results in this research area.Strongly quasinonexpansive mappings. II.https://www.zbmath.org/1456.470162021-04-16T16:22:00+00:00"Aoyama, Koji"https://www.zbmath.org/authors/?q=ai:aoyama.koji"Zembayashi, Kei"https://www.zbmath.org/authors/?q=ai:zembayashi.keiSummary: This paper is devoted to the study of strongly quasinonexpansive mappings in an abstract space and a Banach space.
Editorial remark. Part I has appeared in [\textit{K. Aoyama}, in: Proceedings of the 9th International Conference on Nonlinear analysis and convex analysis, Yokohama: Yokohama Publ. 19--27 (2016; per bibl.)]. Part III has appeared in [\textit{K. Aoyama} and \textit{F. Kohsaka}, Linear Nonlinear Anal. 6, No. 1, 1--12 (2020), \url{http://www.yokohamapublishers.jp/online2/oplna/vol6/p1.html}].The unique periodic solution of Abel's differential equation.https://www.zbmath.org/1456.340422021-04-16T16:22:00+00:00"Hua, Ni"https://www.zbmath.org/authors/?q=ai:hua.niConsider the scalar differential equation
\[
\frac{{dx}}{{dt}} = a(t) x^3 + b(t) x^2 + c(t) x+ d(t) \tag{1}
\]
under the assumption that \(a,b,c,d\) are continuous \(\omega\)-periodic functions. The authors prove that under
the additional conditions
(\(i\)). \( a(t) <0 \quad (a(t)>0)\)
(\(ii\)). \(b^2(t)-3a(t)<0\)
equation (1) has a unique \(\omega\)-periodic solution which is asymptotically stable (unstable).
Reviewer: Klaus R. Schneider (Berlin)Lifschitz tail for alloy-type models driven by the fractional Laplacian.https://www.zbmath.org/1456.601192021-04-16T16:22:00+00:00"Kaleta, Kamil"https://www.zbmath.org/authors/?q=ai:kaleta.kamil"Pietruska-Pałuba, Katarzyna"https://www.zbmath.org/authors/?q=ai:pietruska-paluba.katarzynaSummary: We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators \(( - \Delta )^{\alpha / 2} + V^\omega\) in \(L^2( \mathbb{R}^d)\) for the full range of \(\alpha \in(0, 2]\) and a fairly large class of random nonnegative alloy-type potentials \(V^\omega \). The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit \[\lim_{\lambda \to 0} \lambda^{d / \alpha} \ln N(\lambda) = - C \omega_d \left( \lambda_d^{( \alpha )} \right)^{d / \alpha},\] with \(C \in(0, \infty]\). The constant \(C\) is finite if and only if the common distribution of the lattice random variables charges \(\{0\}\). In this case, the constant \(C\) is expressed explicitly in terms of this distribution. In the limit formula, \( \lambda_d^{( \alpha )}\) denotes the Dirichlet ground-state eigenvalue of the operator \(( - \Delta )^{\alpha / 2}\) in the unit ball in \(\mathbb{R}^d\), and \(\omega_d\) is the volume of this ball.Iterative scheme of strongly nonlinear general nonconvex variational inequalities problem.https://www.zbmath.org/1456.470312021-04-16T16:22:00+00:00"Sudsukh, Chanan"https://www.zbmath.org/authors/?q=ai:sudsukh.chanan"Inchan, Issara"https://www.zbmath.org/authors/?q=ai:inchan.issaraSummary: In this work, we suggest and analyze an iterative scheme for solving strongly nonlinear general nonconvex variational inequalities by using the projection technique and the Wiener-Hopf technique. We prove that strong convergence of the iterative scheme to the solution of the strongly nonlinear general nonconvex variational inequalities requires the modified mapping $T$ to be Lipschitz continuous but not strongly monotone. Our result can be viewed as an improvement of the result of \textit{E. Al-Shemas} [``General nonconvex Wiener-Hopf equations and general nonconvex variational inequalities'', J. Math. Sci., Adv. Appl. 19, No. 1, 1--11 (2013), \url{http://scientificadvances.co.in/admin/img_data/602/images/[1]%20JMSAA%207100121122%20Enab%20Al-Shemas%20[1-11].pdf}].Positive solutions of the \(p\)-Laplacian dynamic equations on time scales with sign changing nonlinearity.https://www.zbmath.org/1456.340842021-04-16T16:22:00+00:00"Dogan, Abdulkadir"https://www.zbmath.org/authors/?q=ai:dogan.abdulkadirSummary: This article concerns the existence of positive solutions for \(p\)-Laplacian boundary value problem on time scales. By applying fixed point index we obtain the existence of solutions. Emphasis is put on the fact that the nonlinear term is allowed to change sign. An example illustrates our results.On ``well posed function spaces'' for \(L^2\)-illposed hyperbolic equations.https://www.zbmath.org/1456.470132021-04-16T16:22:00+00:00"Furuya, Kiyoko"https://www.zbmath.org/authors/?q=ai:furuya.kiyokoSummary: In what kind of function spaces are nonparabolic equations well-posed? This talk is concerned with well-posed function spaces for hyperbolic equations of the simplest type. We consider the mixed problem with oblique boundary condition which is not well-posed in the \(L^2\) sense. First, we introduce the space \(Y\) in which the equation is well-posed. Next, we consider the infinitesimal generator \(A\) which generates a \(C_0\)-semigroup on \(Y\).On the Haugazeau-like projective method for the sum problem.https://www.zbmath.org/1456.470272021-04-16T16:22:00+00:00"Matsushita, Shin-Ya"https://www.zbmath.org/authors/?q=ai:matsushita.shinya"Xu, Li"https://www.zbmath.org/authors/?q=ai:xu.liSummary: The sum problem, which consists in finding zeros of the sum of two maximal monotone operators, provides a powerful framework for modelling a wide variety of real problems. We investigate the basic and important question whether the existence of solutions to the sum problem can be guaranteed. We show that the boundedness of the sequence generated by the Haugazeau-like projective method [\textit{H. Zhang} and \textit{L.-Z. Cheng}, J. Math. Anal. Appl. 406, No. 1, 323--334 (2013; Zbl 1310.47109)] guarantees the existence of solutions to the sum problem.Study of a boundary value problem for fractional order \(\psi\)-Hilfer fractional derivative.https://www.zbmath.org/1456.340052021-04-16T16:22:00+00:00"Harikrishnan, S."https://www.zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Kanagarajan, K."https://www.zbmath.org/authors/?q=ai:kanagarajan.kana|kanagarajan.kuppusamySummary: This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in \(\psi\)-Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.On the minimax inequality of Brézis-Nirenberg-Stampacchia.https://www.zbmath.org/1456.470182021-04-16T16:22:00+00:00"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since the celebrated Knaster-Kuratowski-Mazurkiewicz (simply KKM) theorem appeared in [\textit{B. Knaster} et al., Fundam. Math. 14, 132--137 (1929; JFM 55.0972.01)], a large number of its generalizations and modifications followed. Based on a lemma which generalizes the KKM theorem, Brézis-Nirenberg-Stampacchia (simply BNS) [\textit{H. Brézis} et al., Boll. Unione Mat. Ital., IV. Ser. 6, 293--300 (1972; Zbl 0264.49013); ibid. (9) 1, No. 2, 257--264 (2008; Zbl 1225.49014)] obtained a slightly more general result than the 1961 KKM lemma of \textit{K. Fan} [Math. Ann. 142, 305--310 (1961; Zbl 0093.36701)]. Then they obtained a generalization of the 1972 minimax inequality of \textit{K. Fan} [in: Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 103--113 (1972; Zbl 0302.49019)] and some of its applications.
In the present article, we show that one of our previous KKM type theorems for abstract convex spaces [the author, ``On the von Neumann type minimax theorems in abstract convex spaces'', J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 50, No. 2, 1--24 (2011), \url{http://parksehie.com/datafiles/research/2011L=JNAS.pdf}] can be applied to generalize the KKM type lemma and the minimax inequality due to Brézis-Nirenberg-Stampacchia [loc.\,cit.]. Using our results, we can correct certain results of Brézis-Nirenberg-Stampacchia [loc.\,cit.].Explicit formula and meromorphic extension of the resolvent for the massive Dirac operator in the Schwarzschild-anti-de Sitter spacetime.https://www.zbmath.org/1456.811702021-04-16T16:22:00+00:00"Idelon-Riton, Guillaume"https://www.zbmath.org/authors/?q=ai:idelon-riton.guillaumeSummary: We study the resolvent of the massive Dirac operator in the Schwarzschild-anti-de Sitter space-time. After separation of variables, we use standard one-dimensional techniques to obtain an explicit formula. We then make use of this formula to extend the resolvent meromorphically across the real axis.{
\copyright 2017 American Institute of Physics}Vector-valued functions generated by the operator of finite order and their application to solving operator equations in locally convex spaces.https://www.zbmath.org/1456.470042021-04-16T16:22:00+00:00"Man'ko, Svetlana N."https://www.zbmath.org/authors/?q=ai:manko.svetlana-nSummary: This work is devoted to solving some classes of operator equations, based on the solution of an auxiliary one-parameter family of equations, which is obtained from the original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics and the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.On self-adjoint extensions and symmetries in quantum mechanics.https://www.zbmath.org/1456.811792021-04-16T16:22:00+00:00"Ibort, Alberto"https://www.zbmath.org/authors/?q=ai:ibort.alberto"Lledó, Fernando"https://www.zbmath.org/authors/?q=ai:lledo.fernando"Pérez-Pardo, Juan Manuel"https://www.zbmath.org/authors/?q=ai:perez-pardo.juan-manuelSummary: Given a unitary representation of a Lie group \(G\) on a Hilbert space \(\mathcal H\), we develop the theory of \(G\)-invariant self-adjoint extensions of symmetric operators using both von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the \(G\)-invariant unbounded operator. We also prove a \(G\)-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of \(G\)-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group \(G\) acts. These extensions are labeled by admissible unitaries \(U\) acting on the \(L^2\)-space at the boundary and having spectral gap at \(-1\). It is shown that if the unitary representation \(V\) of the symmetry group \(G\) is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by \(U\) is \(G\)-invariant if \(U\) and \(V\) commute at the boundary. Various significant examples are discussed at the end.Lie triple derivations of incidence algebras.https://www.zbmath.org/1456.160402021-04-16T16:22:00+00:00"Wang, Danni"https://www.zbmath.org/authors/?q=ai:wang.danni"Xiao, Zhankui"https://www.zbmath.org/authors/?q=ai:xiao.zhankuiLet \(A\) be an associative algebra over \(\mathcal{R}\), a commutative ring with unit, and let \(Z(A)\) denote the center of \(A\). An \(\mathcal{R}\)-linear map \(L\colon A\rightarrow A\) is called a Lie triple derivation if \(L([[x,y],z])=[[L(x),y],z]+[[x,L(y)],z]+[[x,y],L(z)]\) for all \(x,y,z\in A\) and where \([x,y]\) denotes the commutator of \(x\), \(y\). A Lie triple derivation is proper if it is of the form \(D+F\), where \(D\colon A\rightarrow A\) is a derivation, and \(F\colon A\rightarrow Z(A)\) is an \(\mathcal{R}\)-linear map.
From now on, \(\mathcal{R}\) denotes a \(2\)-torsion free commutative ring with unit, \(X\) denotes a locally finite preordered set, and \(I(X,\mathcal{R})\) denotes the incidence algebra of \(X\) over \(\mathcal{R}\). In the paper under review, the authors prove that if \(X\) consists of a finite number of connected components, then every Lie triple derivation of \(I(X,\mathcal{R})\) is proper.
Reviewer: Małgorzata E. Hryniewicka (Białystok)Bilinear expansions of lattices of KP \textbf{\( \tau \)}-functions in BKP \textbf{\( \tau \)}-functions: a fermionic approach.https://www.zbmath.org/1456.813042021-04-16T16:22:00+00:00"Harnad, J."https://www.zbmath.org/authors/?q=ai:harnad.john"Orlov, A. Yu."https://www.zbmath.org/authors/?q=ai:orlov.aleksandr-yuSummary: We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) \( \tau \)-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP \(\tau \)-functions, labeled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur \(Q\)-functions. It is deduced using the representations of KP and BKP \(\tau \)-functions as vacuum expectation values (VEVs) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by the insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEVs of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial \(\tau \)-functions of KP and BKP type.
{\copyright 2021 American Institute of Physics}Euler characteristic on noncommutative polyballs.https://www.zbmath.org/1456.460622021-04-16T16:22:00+00:00"Popescu, Gelu"https://www.zbmath.org/authors/?q=ai:popescu.geluSummary: In this paper we introduce and study the Euler characteristic (denoted by \(\chi\)) associated with algebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provide several asymptotic formulas for \(\chi\) and prove some of its basic properties. We show that the Euler characteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of the tensor product of full Fock spaces \(F^2(H_{n_1})\otimes\dots\otimes F^2(H_{n_k})\), and prove that its range coincides with the interval \([0,\infty)\). We obtain an analogue of Arveson's version of the Gauss-Bonnet-Chern Theorem from Riemannian geometry, which connects the curvature to the Euler characteristic. In particular, we prove that if \(\mathcal{M}\) is an invariant subspace of \(F^2(H_{n_1})\otimes\dots\otimes F^2(H_{n_k})\), \(n_i\geq2\), which is graded (generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of the orthocomplement of \(\mathcal{M}\) coincide.Lipschitz estimates for functions of Dirac and Schrödinger operators.https://www.zbmath.org/1456.811742021-04-16T16:22:00+00:00"Skripka, A."https://www.zbmath.org/authors/?q=ai:skripka.a-n|skripka.annaSummary: We establish new Lipschitz-type bounds for functions of operators with noncompact perturbations that produce Schatten class resolvent differences. The results apply to suitable perturbations of Dirac and Schrödinger operators, including some long-range and random potentials, and to important classes of test functions. The key feature of these bounds is an explicit dependence on the Lipschitz seminorm and decay parameters of the respective scalar functions and, in the case of Dirac and Schrödinger operators, on the \(L^p\)- or \(\mathcal{l}^p(L^2)\)-norm of the potential.
{\copyright 2021 American Institute of Physics}Sklyanin-like algebras for \((q\)-)linear grids and \((q\)-)para-Krawtchouk polynomials.https://www.zbmath.org/1456.812652021-04-16T16:22:00+00:00"Bergeron, Geoffroy"https://www.zbmath.org/authors/?q=ai:bergeron.geoffroy"Gaboriaud, Julien"https://www.zbmath.org/authors/?q=ai:gaboriaud.julien"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://www.zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: S-Heun operators on linear and \(q\)-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big \(q\)-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and \(q\)-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators.
{\copyright 2021 American Institute of Physics}Generic zero-Hausdorff and one-packing spectral measures.https://www.zbmath.org/1456.811772021-04-16T16:22:00+00:00"Carvalho, Silas L."https://www.zbmath.org/authors/?q=ai:carvalho.silas-l"de Oliveira, César R."https://www.zbmath.org/authors/?q=ai:de-oliveira.cesar-rSummary: For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneous zero upper-Hausdorff and one lower-packing dimension contains a dense \(G_\delta\) subset. Applications include sets of limit-periodic operators.
{\copyright 2021 American Institute of Physics}Spectral inclusion and pollution for a class of dissipative perturbations.https://www.zbmath.org/1456.811822021-04-16T16:22:00+00:00"Stepanenko, Alexei"https://www.zbmath.org/authors/?q=ai:stepanenko.alexeiSummary: Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form \(T_0 + i \gamma s_n\) on a Hilbert space, where \(s_n\) is strongly convergent to the identity operator and \(\gamma > 0\). We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.
{\copyright 2021 American Institute of Physics}Regular dilation and Nica-covariant representation on right LCM semigroups.https://www.zbmath.org/1456.430042021-04-16T16:22:00+00:00"Li, Boyu"https://www.zbmath.org/authors/?q=ai:li.boyuGeneralizing the celebrated Sz.~Nagy dilation of a single contraction, Brehmer studied regular dilations back in the early sixties. Since then the notion of regular dilation has been investigated by many researchers and has been generalized to product systems, lattice ordered semigroups, and recently to graph products of \(\mathbb{N}\). It was shown by the author of the present paper in [J. Funct. Anal. 273, No. 2, 799--835 (2017; Zbl 06720587)] that for such graph products, the existence of a \(*\)-regular dilation is equivalent to the existence of a minimal isometric Nica-covariant dilation.
In the paper under review, the author extends this result to right LCM semigroups, which are unital left cancellative semigroups \(P\) such that for any \(p, q\in P\), either \(pP\cap qP=\emptyset\) or \(pP\cap qP=rP\) for some \(r\in P\). Such an element \(r\) can be considered as a right least common multiple of \(p\) and \(q\) (hence the name right LCM). The author also shows the equivalence among a \(*\)-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result unifies many previous results on regular dilations, including Brehmer's theorem, Frazho-Bunce-Popescu's dilation of noncommutative row contractions, regular dilations on lattice ordered semigroups and graph products of \(\mathbb{N}\). Applications to many important classes of right LCM semigroups are given. In the last section of the paper, the author obtains a characterization of \(*\)-regular representations of graph products of right LCM semigroups and an application to doubly commuting representations of direct sums of right LCM semigroups.
Reviewer: Trieu Le (Toledo)Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions.https://www.zbmath.org/1456.811802021-04-16T16:22:00+00:00"Lotoreichik, Vladimir"https://www.zbmath.org/authors/?q=ai:lotoreichik.vladimir"Michelangeli, Alessandro"https://www.zbmath.org/authors/?q=ai:michelangeli.alessandroSummary: We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrödinger operator with point interaction: the optimizer is the ball with the point interaction supported at its center. Next, we establish three-dimensional Faber-Krahn inequalities for a one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimizer being given in terms of Coulomb attraction at the center of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model, a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.
{\copyright 2021 American Institute of Physics}Discrete self-adjointness and quantum dynamics. Travel times.https://www.zbmath.org/1456.811732021-04-16T16:22:00+00:00"Martínez-Pérez, Armando"https://www.zbmath.org/authors/?q=ai:martinez-perez.armando"Torres-Vega, Gabino"https://www.zbmath.org/authors/?q=ai:torres-vega.gabinoSummary: We use a discrete derivative to introduce a time operator for non-relativistic quantum systems with point spectrum. The symmetry requirement on the time operator leads to well-defined time values related to the dynamics of discrete quantum systems. As an illustration, we find travel times between hits with the walls for the quantum particle in a box model. These times suggest a classical analog of time eigenstates. We also briefly consider the Woods-Saxon potential and propose classical analogs for it.
{\copyright 2021 American Institute of Physics}Spectral properties of the operators \(AB\) and \(BA\).https://www.zbmath.org/1456.470012021-04-16T16:22:00+00:00"Didenko, D. B."https://www.zbmath.org/authors/?q=ai:didenko.d-bSummary: For linear bounded operators \(A, B\) from the Banach algebra of linear bounded operators acting in a Banach space, we prove a number of statements on the coincidence of the properties of the operators \(I_Y - AB\), \(I_X - BA\) related to their kernels and images. In particular, we establish the identical dimension of the kernels, their simultaneous complementability property, the coincidence of the codimensions of the images, their simultaneous Fredholm property and the coincidence of their Fredholm indices. We construct projections onto the image and the kernel of these operators. We establish the simultaneous nonquasianalyticity property of the operators \(AB\) and \(BA\).