Recent zbMATH articles in MSC 47B https://zbmath.org/atom/cc/47B 2022-06-24T15:10:38.853281Z Werkzeug Inequalities on partial traces of positive semidefinite block matrices https://zbmath.org/1485.15019 2022-06-24T15:10:38.853281Z "Fu, Xiaohui" https://zbmath.org/authors/?q=ai:fu.xiaohui "Lau, Pan-Shun" https://zbmath.org/authors/?q=ai:lau.pan-shun "Tam, Tin-Yau" https://zbmath.org/authors/?q=ai:tam.tin-yau Let $${M_n}$$ and $${M_m}({M_n})$$ be the set of all $$n \times n$$ complex matrices and the set of all $$m \times m$$ block matrices with each block in $${M_n}$$, respectively. Two partial traces of $$A = [{A_{i,j}}]_{i,j =1}^m\in{M_m}({M_n})$$ are defined by ${\text{t}}{{\text{r}}_1}A = \sum\nolimits_{i=1}^m{{A_{i,i}}}\in{M_n},$ and ${\text{t}}{{\text{r}}_2}A=[{\text{tr}}{A_{i,j }}]_{i,j = 1}^m \in {M_m}.$ A density matrix is a positive semidefinite matrix $$A$$ with $${\text{tr A = 1}}$$. For every density matrix $$A \in {M_m}({M_n})$$ a result given by \textit{M. Lin} [Can. Math. Bull. 59, No. 3, 585--591 (2016; Zbl 1359.15013)] shows that $1+\det A\geqslant\det{({\text{t}}{{\text{r}}_1}A)^m} + \det {({\text{t}}{{\text{r}}_2}A)^n}.$ The main results of this paper are the following inequalities, which improve the ones proved by \textit{M. Lin}: $\det ({\text{t}}{{\text{r}}_1}A) \leqslant {(1/n)^n},$ and $\det ({\text{t}}{{\text{r}}_2}A) \leqslant {(1/m)^m}.$ Reviewer: Mihail Voicu (Iaşi) Extensions of some matrix inequalities related to trace and partial traces https://zbmath.org/1485.15025 2022-06-24T15:10:38.853281Z "Li, Yongtao" https://zbmath.org/authors/?q=ai:li.yongtao Summary: We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of \textit{M. Lin} [Czech. Math. J. 66, No. 3, 737--742 (2016; Zbl 1413.15040)] and improves a result of \textit{L. Kuai} [Linear Multilinear Algebra 66, No. 3, 547--553 (2018; Zbl 1427.15022)]. Moreover, we provide a unified treatment of a result of \textit{T. Ando} [Matrix inequalities involving partial traces''; ILAS conference (2014)] and a recent result of \textit{Y. Li} et al. [Oper. Matrices, No. 15, 3, 1189--1199 (2021)]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on trace inequalities for positive semidefinite $$2\times 2$$ block matrices are also included. Norm inequalities for positive semidefinite matrices and a question of Bourin. III https://zbmath.org/1485.15027 2022-06-24T15:10:38.853281Z "Hayajneh, Mostafa" https://zbmath.org/authors/?q=ai:hayajneh.mostafa "Hayajneh, Saja" https://zbmath.org/authors/?q=ai:hayajneh.saja "Kittaneh, Fuad" https://zbmath.org/authors/?q=ai:kittaneh.fuad "Lebaini, Imane" https://zbmath.org/authors/?q=ai:lebaini.imane \textit{J.-C. Bourin} investigated matrix subadditivity inequalities and block matrices in [Int. J. Math. 20, No. 6, 679--691 (2009; Zbl 1181.15030)]. He asked whether $$|||A^tB^{1-t}+B^tA^{1-t}||| \leq |||A+B|||$$ for any unitarily invariant norm $$|||\cdot|||$$ positive semidefinite matrices $$A$$ and $$B$$ and $$t \in [0, 1]$$. \textit{M. Hayajneh} et al. [Int. J. Math. 32, No. 7, Article ID 2150043, 8 p. (2021; Zbl 1475.15026)] answered Bourin's question for the cases $$t=1/4, 1/2, 3/4$$. Among other results concerning weak majorization inequalities, the authors of present paper prove that $\|A^tXB^{1-t}+B^tX^*A^{1-t}\|_p \leq 2^{\frac{1}{2}-\frac{1}{2^p}}\|X\|_\infty\|A+B\|_p$ for positive semidefinite matrices $$A, B$$, and for any matrix $$X$$, $$t\in [0, 1]$$, and $$p \geq 1$$. Reviewer: Mohammad Sal Moslehian (Mashhad) Hyers-Ulam-Rassias stability of $$(m, n)$$-Jordan derivations https://zbmath.org/1485.16038 2022-06-24T15:10:38.853281Z "An, Guangyu" https://zbmath.org/authors/?q=ai:an.guangyu "Yao, Ying" https://zbmath.org/authors/?q=ai:yao.ying.1 Summary: In this paper, we study the Hyers-Ulam-Rassias stability of $$(m, n)$$-Jordan derivations. As applications, we characterize $$(m, n)$$-Jordan derivations on $$C^*$$-algebras and some non-self-adjoint operator algebras. Some Hardy-type inequalities in Banach function spaces https://zbmath.org/1485.26022 2022-06-24T15:10:38.853281Z "Barza, Sorina" https://zbmath.org/authors/?q=ai:barza.sorina "Nikolova, Ljudmila" https://zbmath.org/authors/?q=ai:nikolova.ludmila "Persson, Lars-Erik" https://zbmath.org/authors/?q=ai:persson.lars-erik "Yimer, Markos" https://zbmath.org/authors/?q=ai:yimer.markos In this paper, the authors establish and prove some new Hardy-type inequalities in a Banach function space settings and the results obtained are well discussed and their connections to some well-known results in the literature are well documented and pointed out. The results obtained in this paper further unify and generalize several classical Hardy-type inequalities in the literature. Reviewer: James Adedayo Oguntuase (Abeokuta) Certain properties of multivalent analytic functions defined by $$q$$-difference operator involving the janowski function https://zbmath.org/1485.30008 2022-06-24T15:10:38.853281Z "Wang, Bo" https://zbmath.org/authors/?q=ai:wang.bo.1|wang.bo|wang.bo.2 "Srivastava, Rekha" https://zbmath.org/authors/?q=ai:srivastava.rekha "Liu, Jin-Lin" https://zbmath.org/authors/?q=ai:liu.jinlin Summary: A new subclass of multivalent analytic functions is defined by means of $$q$$-difference operator and Janowski function. Some properties of functions in this new subclass such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radii of starlikeness and convexity, partial sums and closure theorems are studied. Spectral properties of a fourth-order differential operator with eigenvalue parameter-dependent boundary conditions https://zbmath.org/1485.34087 2022-06-24T15:10:38.853281Z "Mehrabov, Vuqar A." https://zbmath.org/authors/?q=ai:mehrabov.vuqar-a Summary: This paper is devoted to the study of the spectral properties of one eigenvalue problem for the fourth-order ordinary differential equations with a spectral parameter contained in two of the boundary conditions. This spectral problem arises when the Fourier method is applied to a boundary value problem for partial differential equations describing small bending vibrations of a homogeneous rod under the action of a longitudinal force in cross sections. The left end of the rod is either free or freely supported, and the inertial mass is concentrated on the right end. We find the arrangement of the eigenvalues on the real axis, determine the multiplicities of all eigenvalues, and study the asymptotic behavior of the eigenvalues and eigenfunctions of this problem. Moreover, sufficient conditions were found under which the system of root functions with two removed functions is a basis in the space $$L_p$$, $$1< p < \infty$$. Regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix https://zbmath.org/1485.35095 2022-06-24T15:10:38.853281Z "Kinzebulatov, D." https://zbmath.org/authors/?q=ai:kinzebulatov.damir "Semënov, Yu. A." https://zbmath.org/authors/?q=ai:semenov.yuliy-a Summary: In $${\mathbb{R}}^d$$, $$d \ge 3$$, consider the divergence and the non-divergence form operators \begin{aligned} & -\Delta - \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \\ & - \Delta - (a-I) \cdot \nabla^2 + b \cdot \nabla, \end{aligned} where the second-order perturbations are given by the matrix $a-I=c|x|^{-2}x \otimes x, \quad c>-1.$ The vector field $$b:{\mathbb{R}}^d \rightarrow{\mathbb{R}}^d$$ is form-bounded with form-bound $$\delta >0$$. (This includes vector fields with entries in $$L^d$$, as well as vector fields having critical-order singularities.) We characterize quantitative dependence on $$c$$ and $$\delta$$ of the $$L^q \rightarrow W^{1,qd/(d-2)}$$ regularity of solutions of the corresponding elliptic and parabolic equations in $$L^q$$, $$q \ge 2 \vee ( d-2)$$. On the nonlinear perturbations of self-adjoint operators https://zbmath.org/1485.35227 2022-06-24T15:10:38.853281Z "Bełdziński, Michał" https://zbmath.org/authors/?q=ai:beldzinski.michal "Galewski, Marek" https://zbmath.org/authors/?q=ai:galewski.marek "Majdak, Witold" https://zbmath.org/authors/?q=ai:majdak.witold Summary: Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation $$Tx=N(x)$$, where $$T$$ is a self-adjoint operator in a real Hilbert space $$\mathcal{H}$$ and $$N$$ is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators. Limiting absorption principle and virtual levels of operators in Banach spaces https://zbmath.org/1485.35300 2022-06-24T15:10:38.853281Z "Boussaid, Nabile" https://zbmath.org/authors/?q=ai:boussaid.nabile "Comech, Andrew" https://zbmath.org/authors/?q=ai:comech.andrew Summary: We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces. Quantitative spectral analysis of electromagnetic scattering. II: Evolution semigroups and non-perturbative solutions https://zbmath.org/1485.35351 2022-06-24T15:10:38.853281Z "Zhou, Yajun" https://zbmath.org/authors/?q=ai:zhou.yajun Summary: We carry out quantitative studies on the Green operator $$\hat{\mathscr{G}}$$ associated with the Born equation, an integral equation that models electromagnetic scattering, building the strong stability of the evolution semigroup $$\{\exp (i\tau G\hat{\mathscr{G}})\vert\tau\geq 0\}$$ on polynomial compactness and the Arendt-Batty-Lyubich-Vũ theorem. The strongly-stable evolution semigroup inspires our proposal of a nonperturbative method to solve the light scattering problem and improve the Born approximation. For Part I, see [Quantitative spectral analysis of electromagnetic scattering. I: $$L^ 2$$ and Hilbert-Schmidt norm bounds'', Preprint, \url{arXiv:1007.4375}]. On a family of Volterra cubic stochastic operators https://zbmath.org/1485.37083 2022-06-24T15:10:38.853281Z "Kurganov, K. A." https://zbmath.org/authors/?q=ai:kurganov.k-a "Jamilov, U. U." https://zbmath.org/authors/?q=ai:jamilov.uygun-u "Okhunova, M. O." https://zbmath.org/authors/?q=ai:okhunova.m-o Summary: In present paper we consider a family of discrete time Kolmogorov systems of three interaction population depending on a parameter $$\theta$$. We show that there is the critic value $$\theta^*$$ of parameter $$\theta$$ such that for $$\theta\in(\theta^*,1]$$ this evolution operator is a non-ergodic transformation and for $$\theta\in[0,\theta^*)$$ it has property being regular. We give some biological interpretations of our results. Properties of critical and subcritical second order self-adjoint linear equations https://zbmath.org/1485.39022 2022-06-24T15:10:38.853281Z "Jekl, Jan" https://zbmath.org/authors/?q=ai:jekl.jan Necessary and/or sufficient conditions guaranteeing criticality (subcriticality) of the equation $$a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}=0$$, $$n\in\mathbb{Z}$$, are established. Further, the fourth order equation $$-\Delta^4 y_n+\Delta(r_{n+1}\Delta y_{n+1})=0$$, $$r_n>0$$, $$n\in\mathbb{Z}$$, is considered, and its 1-criticality is proved. Reviewer: Pavel Rehak (Brno) On nonlinear random approximation of 3-variable Cauchy functional equation https://zbmath.org/1485.39036 2022-06-24T15:10:38.853281Z "Cho, Yeol Je" https://zbmath.org/authors/?q=ai:cho.yeol-je "Kang, Shin Min" https://zbmath.org/authors/?q=ai:kang.shin-min "Rassias, Themistocles M." https://zbmath.org/authors/?q=ai:rassias.themistocles-m "Saadati, Reza" https://zbmath.org/authors/?q=ai:saadati.reza Summary: In this paper, we study to approximate the homomorphisms and derivations for 3-variable Cauchy functional equations in $$RC^\ast$$-algebras and Lie $$RC^\ast$$-algebras by the fixed point method. On the convergence difference sequences and the related operator norms https://zbmath.org/1485.40002 2022-06-24T15:10:38.853281Z "Baliarsingh, P." https://zbmath.org/authors/?q=ai:baliarsingh.pinakadhar "Nayak, L." https://zbmath.org/authors/?q=ai:nayak.laxmipriya "Samantaray, S." https://zbmath.org/authors/?q=ai:samantaray.saurav Summary: In this note, we discuss the definitions of the difference sequences defined earlier by \textit{H.~Kızmaz} [Can. Math. Bull. 24, 169--176 (1981; Zbl 0454.46010)], \textit{M.~Et} and \textit{R.~Çolak} [Soochow J. Math. 21, No.~4, 377--386 (1995; Zbl 0841.46006)], \textit{E.~Malkowsky} et al. [Acta Math. Sin., Engl. Ser. 23, No.~3, 521--532 (2007; Zbl 1123.46007)], \textit{F.~Başar} [Summability theory and its applications. With a foreword by M.~Mursaleen. Oak Park, IL: Bentham Science Publishers (2012; Zbl 1342.40001)], \textit{P.~Baliarsingh} [Appl. Math. Comput. 219, No.~18, 9737--9742 (2013; Zbl 1300.46004)] and many others. Several authors have defined the difference sequence spaces and studied their various properties. It is quite natural to analyze the convergence of the corresponding sequences. As a part of this work, a convergence analysis of difference sequence of fractional order defined earlier is presented. It is demonstrated that the convergence of the fractional difference sequence is dynamic in nature and some of the results involved are also inconsistent. We provide certain stronger conditions on the primary sequence and the results due to earlier authors are substantially modified. Some illustrative examples are provided for each point of the modifications. Results on certain operator norms related to the difference operator of fractional order are also determined. Multiplier conditions for boundedness into Hardy spaces https://zbmath.org/1485.42017 2022-06-24T15:10:38.853281Z "Grafakos, Loukas" https://zbmath.org/authors/?q=ai:grafakos.loukas "Nakamura, Shohei" https://zbmath.org/authors/?q=ai:nakamura.shohei "Nguyen, Hanh Van" https://zbmath.org/authors/?q=ai:van-nguyen.hanh "Sawano, Yoshihiro" https://zbmath.org/authors/?q=ai:sawano.yoshihiro Summary: In the present work we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $$\sigma (\xi_1,\dots,\xi_m)$$ and their derivatives vanish on the hyperplane $$\xi_1+\cdots +\xi_m=0$$. Upper and lower bounds for Littlewood-Paley square functions in the Dunkl Setting https://zbmath.org/1485.42023 2022-06-24T15:10:38.853281Z "Dziubański, Jacek" https://zbmath.org/authors/?q=ai:dziubanski.jacek "Hejna, Agnieszka" https://zbmath.org/authors/?q=ai:hejna.agnieszka In this paper the authors prove some integral bounds for Littlewood-Paley square functions in the Dunkl context. In $$\mathbb R^N$$ endowed with a normalized root system $$R$$ and a multiplicity function $$k\geq 0$$, the authors consider the classical gradient $$\nabla$$, the Dunkl gradient $$\nabla_k$$, the Dunkl Laplacian $$\Delta_k$$ and the carré du champ operator $$\Gamma$$ associated to $$\Delta_k$$. By means of two fixed functions $$\Phi$$ and $$\Psi$$ (not necessarily radial), defined on $$\mathbb R^N$$ and satisfying certain smoothness, integrability and decay conditions, the authors introduce some square functions, which are associated in a natural way to $$\nabla_k$$, $$\Delta_k$$, $$\Gamma$$. By adapting some tecniques from Calderón-Zygmund analysis to the Dunkl framework, they first prove an upper bound for the sum of the $$L^p$$-norms (with respect to a measure, suitably defined in terms of the root system and the multiplicity function) of the square functions associated to $$\nabla$$, $$\nabla_k$$, $$\Gamma$$. Lower bounds for the Dunkl square functions are also proved under an additional condition, stating essentially that the Dunkl transforms of $$\Phi$$ and $$\Psi$$ are not identically zero along any direction. Reviewer: Valentina Casarino (Vicenza) Correction to: On the pillars of functional analysis'' https://zbmath.org/1485.46007 2022-06-24T15:10:38.853281Z "Velasco, M. Victoria" https://zbmath.org/authors/?q=ai:velasco.maria-victoria From the text: Unfortunately, the equation 2.5 in [the author, ibid. 115, No. 4, Paper No. 173, 10 p. (2021; Zbl 1483.46003)] is not updated correctly. Vector lattices with a Hausdorff uo-Lebesgue topology https://zbmath.org/1485.46008 2022-06-24T15:10:38.853281Z "Deng, Yang" https://zbmath.org/authors/?q=ai:deng.yang "de Jeu, Marcel" https://zbmath.org/authors/?q=ai:de-jeu.marcel Summary: We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of $$\operatorname{L}_0(X, \Sigma, \mu)$$ for a semi-finite measure $$\mu$$ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context. Diameter two properties in some vector-valued function spaces https://zbmath.org/1485.46015 2022-06-24T15:10:38.853281Z "Lee, Han Ju" https://zbmath.org/authors/?q=ai:lee.han-ju|lee.han-ju.1 "Tag, Hyung-Joon" https://zbmath.org/authors/?q=ai:tag.hyung-joon Given a locally convex space $$X$$ whose topology is compatible with a dual pair and given a compact Hausdorff space $$K$$, the \textit{$$X$$-valued function space over a uniform algebra}, $$A(K,X)$$, is a closed subspace of $$C(K,X)$$ satisfying \begin{itemize} \item[(a)] $$A:=X^*\circ A(K,X)$$ is a uniform algebra over $$K$$; \item[(b)] $$A\cdot A(K,X)\subset A(K,X)$$. \end{itemize} The paper introduces the class of Banach spaces of the form $$A(K,X)$$ and studies it in order to provide generalizations of examples of spaces enjoying the \textit{diameter two property} (i.e., every relatively weakly open subset of the unit ball has diameter two) and spaces possessing \textit{Daugavet points} (i.e., $$x\in S_X$$ which is such that $$\overline{\text{conv}}\{y:\|y\|\le1$$ and $$\|x-y\|\ge2-\varepsilon\}$$ coincides with the unit ball for every $$\varepsilon>0$$) and \textit{$$\Delta$$-points} (i.e., $$x\in S_X$$ which is contained in $$\overline{\text{conv}}\{y:\|y\|\le1$$ and $$\|x-y\|\ge2-\varepsilon\}$$ for every $$\varepsilon>0$$). The main results can be summarized as follows. {Theorem.} If $$A$$ is infinite-dimensional, then $$A(K,X)$$ has the diameter two property. {Theorem.} If $$A\otimes X\subset A(K,X)$$, then every norm-one $$f\in A(K,X)$$ such that there is a limit point $$t$$ in the Shilov boundary of $$A$$ satisfying $$\|f(t)\|=1$$ is a Daugavet point. If, in addition, $$X$$ is uniformly convex, then the following are equivalent: \begin{itemize} \item[(i)] $$f$$ is a Daugavet point; \item[(ii)] $$f$$ is a $$\Delta$$-point; \item[(iii)] there is a limit point $$t$$ in the Shilov boundary of $$A$$ satisfying $$\|f(t)\|=1$$. \end{itemize} Reviewer: Stefano Ciaci (Tartu) $$L$$-orthogonality in Daugavet centers and narrow operators https://zbmath.org/1485.46016 2022-06-24T15:10:38.853281Z "Rueda Zoca, Abraham" https://zbmath.org/authors/?q=ai:rueda-zoca.abraham Let $$X, Y$$ be Banach spaces, $$L(X, Y)$$ be the space of all bounded linear operators acting from $$X$$ to $$Y$$. A nonzero operator $$G \in L(X, Y)$$ is a \textit{Daugavet center}, if $$\|G + T\| = \|G\| + \|T\|$$ for every $$T \in L(X, Y)$$ of $$\operatorname{rank} T = 1$$. A Banach space $$X$$ has the \textit{Daugavet property} if the identity operator $$\mathrm{Id} \in L(X, Y)$$ is a Daugavet center. An element $$u \in S_{X^{**}}$$ is called \textit{$$L$$-orthogonal} to $$X$$ if $$\|x + u\| = 1 + \|x\|$$ for every $$x \in X$$. In [\textit{A. Rueda Zoca}, Banach J. Math. Anal. 12, No. 1, 68--84 (2018; Zbl 1391.46017)] the following connection between these concepts was established: a separable space $$X$$ has the Daugavet property if and only if the set of elements $$L$$-orthogonal to $$X$$ is $$w^*$$-dense in $$S_{X^{**}}$$. This result on abundance of $L$-orthogonal elements extends to spaces $$X$$ with the Daugavet property of $$\operatorname{dens}(X) =\omega_1$$, but does not extend to spaces of large density characters [\textit{G.~López-Pérez} and \textit{A.~Rueda Zoca}, Adv. Math. 383, Article ID 107719, 17~p. (2021; Zbl 1473.46013)]. The paper under review extends the above results to Daugavet centers and elucidates the role of $$L$$-orthogonal elements for the theory of narrow operators (a class of operators closely related to the Daugavet property). It is shown that if $$\operatorname{dens}(Y) \leqslant \omega_1$$ and $$G \in S_{L(X, Y)}$$ is a Daugavet center with separable range then, for every non-empty reatively $$w^\ast$$-open subset $$W \subset B_{X^{\ast \ast}}$$, its image $$G^{\ast \ast}(W)$$ contains some element $$L$$-orthogonal to $$Y$$. For a separable $$X$$ with the Daugavet property and a narrow operator $$T \in L(X, Y)$$, it is demonstrated that for every $$y \in B_X$$ and non-empty $$w^\ast$$-open subset $$W \subset B_{X^{\ast \ast}}$$ there is an element $$u \in W$$ which is $$L$$-orthogonal to $$X$$ such that $$T^{\ast \ast}(u) = T(y)$$. In the particular case of separable $$T^\ast( Y^\ast)$$ the previous result extends to $$\operatorname{dens}(X) = \omega_1$$, but none of the previous results hold true in large density characters, in particular, there is a counterexample for $$\omega_2$$ under the set-theoretic assumption $$2^c = \omega_2$$. Reviewer: Vladimir Kadets (Kharkiv) Grothendieck-type subsets of Banach lattices https://zbmath.org/1485.46023 2022-06-24T15:10:38.853281Z "Galindo, Pablo" https://zbmath.org/authors/?q=ai:galindo.pablo "Miranda, Vinícius C. C." https://zbmath.org/authors/?q=ai:miranda.vinicius-c-c The paper contains many concepts and results, so it is difficult to give a summary in few words. It contains five sections, where the first is a one page Introduction. In Section~2 almost Grothendieck sets in a Banach lattice $$E$$ are defined as those sets $$A$$ taken to a relatively weakly compact set in $$c_0$$ for every disjoint operator $$T:E\to c_0$$. A main result in this section is that $$B_E$$ is almost Grothendieck iff no disjoint $$T:E\to c_0$$ is onto (Theorem~2.7). In Section~3 almost Grothendieck operators are defined as a weakening of Grothendieck operators. Let $$X$$ be a Banach space and $$E$$ a Banach lattice. A characterization is that $$T:X\to E$$ is almost Grothendieck iff $$T(B_X)$$ is an almost Grothendieck set in~$$E$$. Section~4 is called \textit{Polynomial considerations} while Section~5 is called \textit{Miscellania}. In particular, the relation to some Dunford-Pettis variants is discussed in Section~5. Reviewer: Olav Nygaard (Kristiansand) Fock space on $$\mathbb{C}^\infty$$ and Bose-Fock space https://zbmath.org/1485.46028 2022-06-24T15:10:38.853281Z "Wick, Brett D." https://zbmath.org/authors/?q=ai:wick.brett-d "Wu, Shengkun" https://zbmath.org/authors/?q=ai:wu.shengkun Summary: In this paper, we introduce the Fock space on $$\mathbb{C}^\infty$$ and obtain an isomorphism between the Fock space on $$\mathbb{C}^\infty$$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in [\textit{C.~A. Berger} and \textit{L.~A. Coburn}, J. Funct. Anal. 68, 273--299 (1986; Zbl 0629.47022)]. As a physical application, we study the Gibbs state. Local triple derivations from C*-algebras into their iterated duals https://zbmath.org/1485.46072 2022-06-24T15:10:38.853281Z "Niazi, Mohsen" https://zbmath.org/authors/?q=ai:niazi.mohsen "Miri, Mohammad Reza" https://zbmath.org/authors/?q=ai:miri.mohammad-reza (no abstract) The essential approximate pseudospectrum and related results https://zbmath.org/1485.47006 2022-06-24T15:10:38.853281Z "Ammar, Aymen" https://zbmath.org/authors/?q=ai:ammar.aymen "Jeribi, Aref" https://zbmath.org/authors/?q=ai:jeribi.aref "Mahfoudhi, Kamel" https://zbmath.org/authors/?q=ai:mahfoudhi.kamel Summary: In this paper, we introduce and study the essential approximate pseudospectrum of closed, densely defined linear operators in the Banach space. We begin by the definition and we investigate the characterization, the stability by means of quasi-compact operators and some properties of this pseudospectrum. Compact perturbations resulting in hereditarily polaroid operators https://zbmath.org/1485.47008 2022-06-24T15:10:38.853281Z "Duggal, B. P." https://zbmath.org/authors/?q=ai:duggal.bhagwati-prashad Summary: A Banach space operator $$A\in B({\mathcal X})$$ is polaroid, $$A\in(\mathcal P)$$, if the isolated points of the spectrum $$\sigma(A)$$ are poles of the operator; $$A$$ is hereditarily polaroid, $$A\in(\mathcal {HP})$$, if every restriction of $$A$$ to a closed invariant subspace is polaroid. It is seen that operators $$A\in(\mathcal {HP})$$ have SVEP -- the single-valued extension property -- on $$\Phi_{sf}(A)=\{\lambda: A-\lambda \text{ is semi Fredholm}\}$$. Hence $$\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A): \operatorname{ind}(A-\lambda)>0\}=\varnothing$$ for operators $$A\in(\mathcal {HP})$$, and a necessary and sufficient condition for the perturbation $$A+K$$ of an operator $$A\in B({\mathcal X})$$ by a compact operator $$K \in B({\mathcal X})$$ to be hereditarily polaroid is that $$\Phi_{sf}^+(A)=\varnothing$$. A~sufficient condition for $$A\in B({\mathcal X})$$ to have SVEP on $$\Phi_{sf}(A)$$ is that its component $$\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \operatorname{ind}(A-\lambda)\leq 0\}$$ is connected. We prove: If $$A\in B({\mathcal H})$$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $$K\in B({\mathcal H})$$ such that $$A+K\in(\mathcal {HP})$$ is that $$\Omega_a(A)$$ is connected. On local spectral properties of Hamilton operators https://zbmath.org/1485.47009 2022-06-24T15:10:38.853281Z "Bai, Wurichaihu" https://zbmath.org/authors/?q=ai:bai.wurichaihu "Chen, Alatancang" https://zbmath.org/authors/?q=ai:chen.alatancang Summary: This paper deals with local spectral properties of Hamilton-type operators. The strongly decomposability, Weyl type theorems and hyperinvariant subspace problem of them and the similar properties with their adjoint operators are studied. As corollaries, some local spectral properties of Hamilton operators are obtained. A note on Anderson's theorem in the infinite-dimensional setting https://zbmath.org/1485.47010 2022-06-24T15:10:38.853281Z "Birbonshi, Riddhick" https://zbmath.org/authors/?q=ai:birbonshi.riddhick "Spitkovsky, Ilya M." https://zbmath.org/authors/?q=ai:spitkovsky.ilya-matvey "Srivastava, P. D." https://zbmath.org/authors/?q=ai:srivastava.parmeshwary-dayal Summary: Anderson's theorem states that, if the numerical range $$W(A)$$ of an $$n$$-by-$$n$$ matrix $$A$$ is contained in the unit disk $$\overline{\mathbb{D}}$$ and intersects with the unit circle at more than $$n$$ points, then $$W(A) = \overline{\mathbb{D}}$$. An analogue of this result for compact $$A$$ in an infinite dimensional setting was established by \textit{H.-L. Gau} and \textit{P. Y. Wu} [Proc. Am. Math. Soc. 134, No. 11, 3159--3162 (2006; Zbl 1107.47005)]. We consider here the case of $$A$$ being the sum of a normal and compact operator. Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc https://zbmath.org/1485.47011 2022-06-24T15:10:38.853281Z "Zhou, Xiaoyang" https://zbmath.org/authors/?q=ai:zhou.xiaoyang "Shi, Yanyue" https://zbmath.org/authors/?q=ai:shi.yanyue "Lu, Yufeng" https://zbmath.org/authors/?q=ai:lu.yufeng Summary: In this paper, we study the invariant subspaces and reducing subspace of the weighted Bergman space over polydisc. The minimal reducing subspaces of a class of analytic Toeplitz operators are completely described, and Beurling-type theorem of some invariant subspaces of Toeplitz operators $$T_{z_i}$$ ($$1\leqslant i\leqslant n$$) on the weighted Bergman space over polydisc is also obtained. Li-Yorke chaos for composition operators on $$L^p$$-spaces https://zbmath.org/1485.47012 2022-06-24T15:10:38.853281Z "Bernardes, N. C. jun." https://zbmath.org/authors/?q=ai:bernardes.nilson-c-jun "Darji, U. B." https://zbmath.org/authors/?q=ai:darji.udayan-b "Pires, B." https://zbmath.org/authors/?q=ai:pires.benito Let $$(X,\mathcal{B},\mu)$$ be a measure space and $$f:X\to X$$ be measurable such that $$T_f:\phi\to \phi\circ f$$ is a bounded linear operator on $$L^p(X,\mathcal{B},\mu)$$ with $$1\leq p<\infty$$. The main result of this paper is a characterization of Li-Yorke chaos for such composition operators $$T_f$$ on $$L^p(X,\mathcal{B},\mu)$$. Several consequences are obtained when $$\mu$$ and/or the symbol $$f$$ have some extra properties. As a consequence, a~characterization of Li-Yorke chaos in the particular case of weighted shifts on $$\ell^p(\mathbb{Z})$$ is obtained. The article closes with several particular examples and counterexamples proving the optimality of some results. Reviewer: Romuald Ernst (Calais) Singular value inequalities for Hilbert space operators https://zbmath.org/1485.47013 2022-06-24T15:10:38.853281Z "Alfakhr, Mahdi Taleb" https://zbmath.org/authors/?q=ai:alfakhr.mahdi-taleb "Omidvar, Mohsen Erfanian" https://zbmath.org/authors/?q=ai:omidvar.mohsen-erfanian Summary: In this paper, we show that, if $$A_i, B_i, X_i$$ are Hilbert space operators such that $$X_i$$ is compact $$i=1,2, \dots,n$$ and $$f, g$$ are non-negative continuous functions on $$[0, \infty)$$ with $$f(t)g(t)=t$$ for all $$t\in[0,\infty)$$, and $$h$$ is a non-negative increasing operator convex function on $$[0, \infty)$$, then $h\left(s_j\left(\sum^n_{i=1} \omega_iA_i^\ast X_i^\ast B_i\right)\right) \leq s_j \left(\sum^n_{i=1} \omega_i h(A_i^\ast f(|X_i^\ast|)^2A_i) \oplus \sum^n_{i=1} \omega_ih(B_i^\ast g(|X_i|)^2B_i)\right)$ for $$j = 1,2,\dots$$ and $$\sum^n_{i=1} \omega_i = 1$$. Also, applications of some inequalities are given. Singular integrals, rank one perturbations and Clark model in general situation https://zbmath.org/1485.47016 2022-06-24T15:10:38.853281Z "Liaw, Constanze" https://zbmath.org/authors/?q=ai:liaw.constanze "Treil, Sergei" https://zbmath.org/authors/?q=ai:treil.sergei Summary: We start with considering rank one self-adjoint perturbations $$A_{\alpha}=A+ \alpha (\cdot, \varphi) \varphi$$ with cyclic vector $$\varphi \in \mathcal{H}$$ on a separable Hilbert space $$\mathcal{H}$$. The spectral representation of the perturbed operator $$A_{\alpha}$$ is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators $$A$$ and $$A_{\alpha}$$. Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle. This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings. Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role. Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators. These lecture notes give an account of the mini-course delivered by the authors, which was centered around [\textit{C. Liaw} and \textit{S. Treil}, J. Funct. Anal. 257, No. 6, 1947--1975 (2009; Zbl 1206.42012); Rev. Mat. Iberoam. 29, No. 1, 53--74 (2013; Zbl 1272.42011); J. Anal. Math. 130, 287--328 (2016; Zbl 06697868)]. Unpublished results are restricted to the last part of this manuscript. For the entire collection see [Zbl 1381.00044]. On properties of the operator equation $$TT^\ast = T + T^\ast$$ https://zbmath.org/1485.47018 2022-06-24T15:10:38.853281Z "An, Il Ju" https://zbmath.org/authors/?q=ai:an.il-ju "Ko, Eungil" https://zbmath.org/authors/?q=ai:ko.eungil Summary: In this paper, we study properties of the operator equation $$TT^\ast =T+T^\ast$$ which \textit{T. T. West} observed in [Proc. Glasg. Math. Assoc. 7, 34--38 (1965; Zbl 0128.35102)]. We first investigate the structure of solutions $$T \in B(\mathcal{H})$$ of such equation. Moreover, we prove that if $$T$$ is a polynomial root of solutions of that operator equation, then the spectral mapping theorem holds for Weyl and essential approximate point spectra of $$T$$ and $$f(T)$$ satisfies $$a$$-Weyl's theorem for $$f \in H(\sigma(T))$$, where $$H(\sigma(T))$$ is the space of functions analytic in an open neighborhood of $$\sigma (T)$$. Some irreducibilities of operators on Banach spaces https://zbmath.org/1485.47023 2022-06-24T15:10:38.853281Z "Zhang, Yunnan" https://zbmath.org/authors/?q=ai:zhang.yunnan "Zhong, Huaijie" https://zbmath.org/authors/?q=ai:zhong.huaijie "Lin, Liqiong" https://zbmath.org/authors/?q=ai:lin.liqiong Summary: According to the definition of Cowen-Douglas operators, this paper gives the concepts of two classes of operators, $$\mathcal{B}_n$$ operators and $$\mathcal{B}$$ operators, which are closely related to strongly irreducible operators. It shows that there exist $$\mathcal{B}_n$$ operators and $$\mathcal{B}$$ operators on separable Banach spaces. It discusses in detail the relationships among the classes of operators with irreducibility and obtains a relationship diagram. This paper also gives some properties of these classes of operators, such as the (quasi) similar invariance of operators. Small operator ideals formed by $$s$$ numbers on generalized Cesàro and Orlicz sequence spaces https://zbmath.org/1485.47024 2022-06-24T15:10:38.853281Z "Faried, Nashat" https://zbmath.org/authors/?q=ai:faried.nashat "Bakery, Awad A." https://zbmath.org/authors/?q=ai:bakery.awad-a Summary: In this article, we establish sufficient conditions on the generalized Cesàro and Orlicz sequence spaces $$\mathbb{E}$$ such that the class $$S_{\mathbb{E}}$$ of all bounded linear operators between arbitrary Banach spaces with its sequence of $$s$$-numbers belonging to $$\mathbb{E}$$ generates an operator ideal. The components of $$S_{\mathbb{E}}$$ as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. Some inclusion relations between the operator ideals as well as the inclusion relations for their duals are obtained. Finally, we show that the operator ideal formed by $$\mathbb{E}$$ and approximation numbers is small under certain conditions. Approximation of mixed order Sobolev functions on the $$d$$-torus: asymptotics, preasymptotics, and $$d$$-dependence https://zbmath.org/1485.47025 2022-06-24T15:10:38.853281Z "Kühn, Thomas" https://zbmath.org/authors/?q=ai:kuhn.thomas "Sickel, Winfried" https://zbmath.org/authors/?q=ai:sickel.winfried "Ullrich, Tino" https://zbmath.org/authors/?q=ai:ullrich.tino Summary: We investigate the approximation of $$d$$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ on the $$d$$-dimensional torus, where the approximation error is measured in the $$L_2$$-norm. In other words, we study the approximation numbers $$a_n$$ of the Sobolev embeddings $$H^s_{\mathrm{mix}}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$$, with particular emphasis on the dependence on the dimension $$d$$. For any fixed smoothness $$s>0$$, we find two-sided estimates for the approximation numbers as a function in $$n$$ and $$d$$. We observe super-exponential decay of the constants in $$d$$, if $$n$$, the number of linear samples of $$f$$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small $$n$$). We present some surprising results for the so-called preasymptotic'' decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems. On products of nuclear operators https://zbmath.org/1485.47026 2022-06-24T15:10:38.853281Z "Reinov, O. I." https://zbmath.org/authors/?q=ai:reinov.oleg-i Summary: The possibility of factoring a product of nuclear operators through operators in the von Neumann-Schatten class is considered. In particular, generally, the product of two nuclear operators can be factored only through a Hilbert-Schmidt operator. Spectral characterizations of infimum and supremum cosine angle between two closed subspaces https://zbmath.org/1485.47027 2022-06-24T15:10:38.853281Z "Wang, Yueqing" https://zbmath.org/authors/?q=ai:wang.yue-qing "Du, Hongke" https://zbmath.org/authors/?q=ai:du.hongke "Zuo, Ning" https://zbmath.org/authors/?q=ai:zuo.ning Summary: In this note, using the technique of block-operators, the spectral characterizations for the infimum cosine angle and the supremum cosine angle between two closed subspaces of $$\mathcal{H}$$ are obtained. Some related results are established. $$n$$-power-posinormal operators https://zbmath.org/1485.47028 2022-06-24T15:10:38.853281Z "Beiba, El Moctar Ould" https://zbmath.org/authors/?q=ai:beiba.el-moctar-ould An operator $$T$$ on a Hilbert space is called posinormal if there exists such a positive operator $$P$$ that $$[T^*,T]=T^*(I-P)T$$ (see [\textit{H. Crawford Rhaly jun.}, J. Math. Soc. Japan 46, No. 4, 587--605 (1994; Zbl 0820.47027)]). A~square of an operator of this class need not be posinormal. The author introduces a class of operators invariant under natural powers and containing all natural powers of any posinormal operator. Reviewer: Anatoly N. Kochubei (Kyïv) Class of $$(A,n)$$-power-hyponormal operators in semi-Hilbertian space https://zbmath.org/1485.47029 2022-06-24T15:10:38.853281Z "Chellali, Cherifa" https://zbmath.org/authors/?q=ai:chellali.cherifa "Benali, Abdelkader" https://zbmath.org/authors/?q=ai:benali.abdelkader In [`On $$n$$-power-hyponormal operators'', Global J. Pure Applied Math. 12, No. 1, 473--479 (2016), \url{https://www.ripublication.com/gjpam16/gjpamv12n1_45.pdf}], \textit{M. Guesba} and \textit{M. Nadir} introduced the concept of $$n$$-power hyponormal operators on a Hilbert space. In the paper under review, this concept is generalized when an additional semi-inner product is considered and some interesting properties of $$(A,n)$$-power-hyponormal operators in semi-Hilbertian spaces are presented. Reviewer: Tudor Bînzar (Timişoara) Generalized Weyl's theorem and spectral continuity for $$(n, k)$$-quasiparanormal operators https://zbmath.org/1485.47030 2022-06-24T15:10:38.853281Z "Gao, Fugen" https://zbmath.org/authors/?q=ai:gao.fugen "Zhang, Qian" https://zbmath.org/authors/?q=ai:zhang.qian Summary: If $$T$$ or $$T^*$$ is a totally $$(n, k)$$-quasiparanormal operator acting on an infinite dimensional separable Hilbert space, then we prove that generalized Weyl's theorem holds for $$f(T)$$ for every $$f\in H (\sigma (T))$$ which is nonconstant on each connected component of its domain. Moreover, if $$T^*$$ is a totally $$(n, k)$$-quasiparanormal operator, then generalized $$a$$-Weyl's theorem holds for $$f(T)$$ for every $$f \in H (\sigma (T))$$ which is nonconstant on each connected component of its domain. Also, we prove that the spectrum is continuous on the class of all $$(n, k)$$-quasiparanormal operators. Tensor product and variants of Weyl's type theorem for $$p$$-$$w$$-hyponormal operators https://zbmath.org/1485.47031 2022-06-24T15:10:38.853281Z "Rashid, M. H. M." https://zbmath.org/authors/?q=ai:rashid.malik-h-m|rashid.mohammad-hussein-mohammad An operator $$T$$ on a Hilbert space is called $$p$$-$$w$$-hyponormal if $$\vert \tilde{T}\vert ^p\ge \vert T\vert ^p\ge \vert \tilde{T}^*\vert ^p$$ where $$\tilde{T}$$ is the Aluthge transform, that is $$\tilde{T}=\vert T\vert ^{1/2}U\vert T\vert ^{1/2}$$ where $$U$$ is the partial isometry appearing in the polar decomposition of $$T$$. The author studies properties of operators from the above class. Reviewer: Anatoly N. Kochubei (Kyïv) Adjoint and unitary operators on Hilbert spaces https://zbmath.org/1485.47032 2022-06-24T15:10:38.853281Z "Hassan, Eltigani Ismail" https://zbmath.org/authors/?q=ai:hassan.eltigani-ismail Some basic concepts and results are recalled. Some special expanded unitary operators on a Hilbert space are described and discussed. It is shown that certain bounded linear operators on a Hilbert space are unitary operators, and also that a Sturm-Liouville operator is self-adjoint. Reviewer: Kui Ji (Shijiazhuang) Semigroup generation properties of Hamiltonian operator matrices https://zbmath.org/1485.47033 2022-06-24T15:10:38.853281Z "Liu, Jie" https://zbmath.org/authors/?q=ai:liu.jie.7|liu.jie.3|liu.jie.1|liu.jie|liu.jie.5|liu.jie.4|liu.jie.2 "Huang, Junjie" https://zbmath.org/authors/?q=ai:huang.junjie "Alatancang" https://zbmath.org/authors/?q=ai:chen.alatancang Summary: This paper deals with the problem for Hamiltonian operator matrices to generate contraction semigroups. The dissipativity of the operator matrix is characterized by those of its diagonal entries. By means of space decomposition and quadratic complements, it is described that the right real axis is contained in the resolvent set of the operator matrix. Based on these properties, some necessary and sufficient conditions on semigroup generation are further given. Essential self-adjointness of anticommutative operators https://zbmath.org/1485.47034 2022-06-24T15:10:38.853281Z "Takaesu, Toshimitsu" https://zbmath.org/authors/?q=ai:takaesu.toshimitsu Summary: The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered. Compact linear combination of composition operators on Bergman spaces https://zbmath.org/1485.47035 2022-06-24T15:10:38.853281Z "Choe, Boo Rim" https://zbmath.org/authors/?q=ai:choe.boo-rim "Koo, Hyungwoon" https://zbmath.org/authors/?q=ai:koo.hyungwoon "Wang, Maofa" https://zbmath.org/authors/?q=ai:wang.maofa Authors' abstract: Motivated by the question of Shapiro and Sundberg raised in [\textit{J. H. Shapiro} and \textit{C. Sundberg}, Pac. J. Math. 145, No. 1, 117--152 (1990; Zbl 0732.30027)], the study of linear combinations of composition operators has been a topic of growing interest. In this paper, we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted Bergman spaces in two classical terms: one is a function theoretic characterization of Julia-Carathéodory type and the other is a measure theoretic characterization of Carleson type. Our approach is completely different from what has been known so far. Reviewer: Raymond Mortini (Metz) Sparse domination of weighted composition operators on weighted Bergman spaces https://zbmath.org/1485.47036 2022-06-24T15:10:38.853281Z "Hu, Bingyang" https://zbmath.org/authors/?q=ai:hu.bingyang "Li, Songxiao" https://zbmath.org/authors/?q=ai:li.songxiao "Shi, Yecheng" https://zbmath.org/authors/?q=ai:shi.yecheng "Wick, Brett D." https://zbmath.org/authors/?q=ai:wick.brett-d The paper uses the technique of sparse domination from harmonic analysis to study several problems for the (holomorphic) Bergman spaces of the upper-half plane and the open unit ball. The problems studied include Carleson embedding, the boundedness and compactness of weighted composition operators, and weighted type estimates for functions in the holomorphic Bergman spaces. Reviewer: Kehe Zhu (Albany) Fredholm generalized composition operators on weighted Hardy spaces https://zbmath.org/1485.47037 2022-06-24T15:10:38.853281Z "Sharma, Sunil Kumar" https://zbmath.org/authors/?q=ai:sharma.sunil-kumar "Gandhi, Rohit" https://zbmath.org/authors/?q=ai:gandhi.rohit "Komal, B. S." https://zbmath.org/authors/?q=ai:komal.b-s Summary: The main purpose of this paper is to study Fredholm generalized composition operators on weighted Hardy spaces. The composition type operator on the weighted Bergman space in the unit ball https://zbmath.org/1485.47038 2022-06-24T15:10:38.853281Z "Zhang, Xuejun" https://zbmath.org/authors/?q=ai:zhang.xuejun "Xiao, Jianbin" https://zbmath.org/authors/?q=ai:xiao.jianbin "Li, Min" https://zbmath.org/authors/?q=ai:li.min.1|li.min.9|li.min.2|li.min.4|li.min.5|li.min.8|li.min|li.min.7|li.min.3|li.min.6|li.min.10 "Guan, Ying" https://zbmath.org/authors/?q=ai:guan.ying Summary: In this paper, necessary and sufficient conditions are given for the weighted composition operator $$T_{\psi, \varphi}$$ to be bounded (or compact) from the weighted Bergman space $$A^p_\alpha$$ to the Bloch type space $$\beta^q$$ on the unit ball in $$\mathbb{C}^n$$. At the same time, the following results are given: (1) If composition operator $$C_\varphi$$ is bounded on $$A^p_\alpha$$, then $$C_\varphi$$ is a compact operator on $$A^p_\alpha$$ if and only if $$\lim_{|z|\to 1}-\frac {1-|z|}{1-|\varphi(z)|}=0$$. The result improves the corresponding result in literature. (2) Composition operator $$C_\varphi$$ is a compact operator from $$A^p_\alpha$$ to $$\beta^{\frac {n+1+\alpha +p}{p}}$$ if and only if $$\lim_{|z|\to 1}-\frac {1-|z|}{1-|\varphi(z)|}=0$$. Isometric weighted composition operators on weighted Bergman spaces https://zbmath.org/1485.47039 2022-06-24T15:10:38.853281Z "Zorboska, Nina" https://zbmath.org/authors/?q=ai:zorboska.nina Summary: We characterize the isometric weighted composition operators on weighted Bergman spaces over the unit disk. We also determine the Wold decomposition of isometric weighted composition operators acting on a class of general reproducing kernel Hilbert spaces in the case when the composition symbol has an interior fixed point, and characterize the numerical range of isometric weighted composition operators on weighted Bergman spaces. Fredholm Toeplitz operators on the weighted Bergman spaces https://zbmath.org/1485.47040 2022-06-24T15:10:38.853281Z "Das, Namita" https://zbmath.org/authors/?q=ai:das.namita "Roy, Swarupa" https://zbmath.org/authors/?q=ai:roy.swarupa Summary: In this paper, we have shown that if $$\phi\in (L_h^2(dA_\alpha))^\bot\cap L^\infty(\mathbb{D})$$ and $$\operatorname{Range}T_\phi^{(\alpha)}$$ is closed, then the Toeplitz operator $$T_\phi^{(\alpha)}\in \mathcal{L} (L_a^2(dA_\alpha))$$ is a~Fredholm operator of index zero and $$T_\phi^{(\alpha)}$$ is not of finite rank. Several applications of the result were also obtained. We further show that if $$\phi\in L_{M_n}^\infty(\mathbb{D})$$ is such that $$T_\phi$$ is Fredholm and of index zero in $$\mathcal{L}(L_a^{2,\mathbb{C}^n}(dA_\alpha))$$, then there exists $$\psi\in E_{n\times n} = E\otimes M_n$$ such that $$T_{\phi+\delta\phi}$$ is invertible for all sufficiently small nonzero $$\delta$$. Here, $$E$$ is a total subspace of $$L^\infty(\mathbb{D})$$ and $$M_n$$ is the set of all $$n\times n$$ matrices with complex entries. Dual truncated Toeplitz operators https://zbmath.org/1485.47041 2022-06-24T15:10:38.853281Z "Ding, Xuanhao" https://zbmath.org/authors/?q=ai:ding.xuanhao "Sang, Yuanqi" https://zbmath.org/authors/?q=ai:sang.yuanqi Summary: Let $$u$$ be a nonconstant inner function. In this paper, we study the dual truncated Toeplitz operators on the orthogonal complement of the model space $$K_u^2$$. This is a new class of Toeplitz operator. We show the product of two dual truncated Toeplitz operators $$D_f D_g$$ to be zero if and only if either $$f$$ or $$g$$ is zero. We give a necessary and sufficient condition for the product of two dual truncated Toeplitz operators to be a finite rank operator. Furthermore, a necessary and sufficient condition is found for the product of two dual truncated Toeplitz operators to be a dual truncated Toeplitz operator. The last two results are different from the classical Toeplitz operator theory. Separately quasihomogeneous Toeplitz operators on the Bergman space of the polydisk https://zbmath.org/1485.47042 2022-06-24T15:10:38.853281Z "Dong, Xingtang" https://zbmath.org/authors/?q=ai:dong.xingtang "Zhou, Zehua" https://zbmath.org/authors/?q=ai:zhou.zehua Summary: The paper is devoted to the study of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the polydisk. First, we obtain necessary and sufficient conditions for the product of two Toeplitz operators with separately quasihomogeneous symbols to be a Toeplitz operator. Next, we provide a decomposition of $$L^2(D^n, dV)$$. Then we use this to show that the zero product of two Toeplitz operators has only one trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Also, the corresponding commuting problem of Toeplitz operators is studied. Commuting Toeplitz operators on the Dirichlet space of the ploydisk https://zbmath.org/1485.47043 2022-06-24T15:10:38.853281Z "Geng, Ligang" https://zbmath.org/authors/?q=ai:geng.ligang "Zhou, Zehua" https://zbmath.org/authors/?q=ai:zhou.zehua Summary: In this paper, we completely characterize the normal Toeplitz operator and commutator of two Toeplitz operators with conjugate holomorphic or holomorphic symbols on the Dirichlet space of the polydisk. We show that the two Toeplitz operators with conjugate holomorphic symbols are commutative on the Dirichlet space of polydisk if and only if two symbols are linearly dependent, and also prove that Toeplitz operator with holomorphic symbol and Toeplitz operator with conjugate holomorphic symbol are commutative if and only if one of the two symbols is a constant. Fredholm properties of Toeplitz operators on the Hardy space $$H^1(S)$$ https://zbmath.org/1485.47044 2022-06-24T15:10:38.853281Z "Huang, Sui" https://zbmath.org/authors/?q=ai:huang.sui "Cao, Guangfu" https://zbmath.org/authors/?q=ai:cao.guangfu Summary: In this paper, we study the Fredholm properties of Toeplitz operators on $$H^1(S)$$. We prove that if a nowhere vanishing symbol function $$\varphi$$ satisfying a slightly stronger condition than $$\varphi$$ is continuous on $$S$$, then $$T_\varphi$$ is Fredholm on $$H^1(S)$$, where $$S$$ is the unit sphere of $$C^n$$. Moreover, when $$n>1$$, the index of $$T_\varphi$$ is zero. Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents https://zbmath.org/1485.47045 2022-06-24T15:10:38.853281Z "Hu, Zhangjian" https://zbmath.org/authors/?q=ai:hu.zhangjian "Virtanen, Jani A." https://zbmath.org/authors/?q=ai:virtanen.jani-a Let $$\omega_0=dd^c|z|^2$$ be the Euclidean Kähler form on the $$n$$-dimensional complex Euclidean space $$\mathbb{C}^n$$, where $$d^c=\frac{\sqrt{-1}}{4}(\overline{\partial}-\partial)$$. Let $$\varphi$$ be a real-valued function in $$C^2(\mathbb{C}^n)$$ such that $$dd^c\varphi$$ is comparable with $$\omega_0$$. For $$p\in(0,\infty)$$, the space $$L^p_\varphi$$ consists of all Lebesgue measurable functions $$f$$ on $$\mathbb{C}^n$$ for which $\big\|f\big\|_{p,\varphi}=\bigg(\int_{\mathbb{C}^n}\big|f(z) e^{-\varphi(z)}\big|^p dv(z)\bigg)^{1/p}<\infty,$ where $$v$$ is the Lebesgue volume measure on $$\mathbb{C}^n$$. For $$p\in(0,\infty)$$, the generalized Fock space with weight $$\varphi$$ is defined by $$F^p_\varphi=L^p_\varphi\cap H(\mathbb{C}^n)$$, where $$H(\mathbb{C}^n)$$ is the set of all holomorphic functions on $$\mathbb{C}^n$$. Let $$\mathrm{VMO}$$ denote the space of vanishing mean oscillation functions on $$\mathbb{C}^n$$. The main result reads as follows: the Toeplitz operator $$T_f$$ with a symbol $$f\in \mathrm{VMO}$$ is Fredholm on the generalized Fock space $$F^p_\varphi$$ with $$p\in(0,\infty)$$ if and only if $0<\liminf_{|z|\to\infty}\big|\widetilde{f}(z)\big|\le \limsup_{|z|\to\infty}\big|\widetilde{f}(z)\big|<\infty,$ where $$\widetilde{f}$$ is the Berezin transform of $$f$$. Reviewer: Yuri I. Karlovich (Cuernavaca) Toeplitz operators on the pluriharmonic Hardy space https://zbmath.org/1485.47046 2022-06-24T15:10:38.853281Z "Liu, Yuan" https://zbmath.org/authors/?q=ai:liu.yuan "Ding, Xuanhao" https://zbmath.org/authors/?q=ai:ding.xuanhao Summary: Corresponding to harmonic Bergman space, in this paper we study Toeplitz operators of pluriharmonic Hardy space. We find that the properties of Toeplitz operators on $$h^2(D^2)$$ are greatly different from the classical Hardy space, Bergman space and harmonic Bergman space. For example, two analytic Toeplitz operators may be not semi-commutable or commutable. Even if it is semi-commutable, neither of both symbols is constant. Even if it is commutable, their symbols' nontrivial linear combination may be not constant. We obtain the necessary and sufficient conditions under which the two analytic Toeplitz operators are semi-commutable and commutable on $$h^2(D^2)$$. The commutant and similarity of analytic Toeplitz operators on weighted Bergman space https://zbmath.org/1485.47047 2022-06-24T15:10:38.853281Z "Li, Yucheng" https://zbmath.org/authors/?q=ai:li.yucheng "Liu, Haifeng" https://zbmath.org/authors/?q=ai:liu.haifeng "Li, Xiangye" https://zbmath.org/authors/?q=ai:li.xiangye Summary: In this paper, we prove that the analytic Toeplitz operator which is induced by $$n$$-Blaschke factors is similar to $$n$$ copies of the Bergman shift on the weighted Bergman space $$A^2_\alpha(\mathbb{D})$$ ($$\alpha>-1$$). We also characterize the commutant of a class of analytic Toeplitz operators and research the similarity of two analytic Toeplitz operators. On Toeplitz operators with quasi-radial and pseudo-homogeneous symbols https://zbmath.org/1485.47048 2022-06-24T15:10:38.853281Z "Vasilevski, Nikolai" https://zbmath.org/authors/?q=ai:vasilevski.nikolai-l From the text: We explore a new wide class of symbols that generate commutative Banach algebras on each weighted Bergman space on the unit ball in $$\mathbb{C}^{n}$$. These symbols are a natural extension of the previously studied quasi-radial quasi-homogeneous symbols [\textit{N. Vasilevski}, Integral Equations Oper. Theory 66, No. 1, 141--152 (2010; Zbl 1216.47050)]. For the case of the two-dimensional ball $$\mathbb{B}^{2}$$, the much wider class of quasi-homogeneous like symbols was described in [\textit{A. García} and \textit{N. Vasilevski}, J. Funct. Spaces 2015, Article ID 306168, 10 p. (2015; Zbl 1321.47158)]. These symbols also generate via corresponding Toeplitz operators commutative Banach algebras on each weighted Bergman space, and their existence was hidden just by the use of the spherical coordinates in the approach of Vasilevski [loc. cit.]. This paper extends the results of García and Vasilevski [loc. cit.] for the case of the unit ball $$\mathbb{B}^{n}$$ with $$n > 2$$, and explores a new wide class of symbols that generate commutative Banach algebras on each weighted Bergman space on these balls. We call these new symbols pseudo-homogeneous, they include the previous quasi-homogeneous symbols as a very special particular case. Roughly speaking, instead of a fixed specific bounded continuous function we admit now any $$L_{\infty}$$-function. For the entire collection see [Zbl 1381.00044]. Some properties of Toeplitz operators on Dirichlet space with respect to Berezin type transform https://zbmath.org/1485.47049 2022-06-24T15:10:38.853281Z "Wang, Xiaofeng" https://zbmath.org/authors/?q=ai:wang.xiaofeng.1 "Xia, Jin" https://zbmath.org/authors/?q=ai:xia.jin "Cao, Guangfu" https://zbmath.org/authors/?q=ai:cao.guangfu Summary: In this paper, we consider the Toeplitz operators and little Hankel operators on the Dirichlet space $$\mathcal{D}$$ of the unit disc $$\mathbb{D}$$. By Berezin type transform, the invariant subspaces of Toeplitz operators, asymptotic multiplicative property of the Berezin type symbols of Toeplitz operators and solvability of the Riccati equations of Toeplitz operators are discussed. We also obtain a sufficient condition for invertibility of Toeplitz operators and little Hankel operators in terms of Berezin transform. Moreover, the Hankel operators and Berezin transform are used to characterize compactness of operators $$2T_{uv}-T_uT_v-T_vT_u$$ for functions $$u$$ and $$v$$ in $$L^{2,1}$$. Commutators and semi-commutators of monomial Toeplitz operators on the pluriharmonic Hardy space https://zbmath.org/1485.47050 2022-06-24T15:10:38.853281Z "Zhang, Yingying" https://zbmath.org/authors/?q=ai:zhang.yingying.1|zhang.yingying.3|zhang.yingying.4|zhang.yingying.2 "Dong, Xingtang" https://zbmath.org/authors/?q=ai:dong.xingtang For $$n\in\mathbb{N}$$, let $$d\mu$$ and $$d\sigma$$ denote, respectively, the normalized Haar measure on the torus $$\mathbb{T}^n$$ being the Cartesian product of $$n$$ copies of the unit circle $$\mathbb{T}\subset \mathbb{C}$$ and the surface area measure on the unit sphere $$\mathbb{S}_n$$. The pluriharmonic Hardy space $$h^2(\mathbb{T}^n)$$ is defined by $$h^2(\mathbb{T}^n)=H^2(\mathbb{T}^n)+\overline{H^2(\mathbb{T}^n)}$$, where the Hardy space $$H^2(\mathbb{T}^n)$$ is the closure of the set of analytic polynomials in the space $$L^2(\mathbb{T}^n,d\mu)$$. Similarly, the pluriharmonic Hardy space $$h^2(\mathbb{S}_n)$$ is the closure of the set of all pluriharmonic functions in the space $$L^2(\mathbb{S}_n,d\sigma)$$. Let $$\Omega_n$$ be $$\mathbb{T}^n$$ or $$\mathbb{S}_n$$. The Toeplitz operator $$T_f$$ with symbol $$f\in L^\infty(\Omega_n)$$ on the pluriharmonic Hardy space $$h^2(\Omega_n)$$ is defined by $$T_f(h)=Q(fh)$$ for all $$h\in h^2(\Omega_n)$$, where $$Q$$ is the orthogonal projection of $$L^2(\Omega_n)$$ onto $$h^2(\Omega_n)$$. The paper gives a complete characterization of the finite rank commutators and semi-commutators of two Toeplitz operators $$T_{f_1}$$ and $$T_{f_2}$$ with monomial symbols of the form $$f=z^p\bar{z}^q$$ on the space $$h^2(\Omega_n)$$, where $$z=(z_1,z_2,\dots,z_n)$$, $$z^p=z_1^{p_1}z_2^{p_2}\dots z_n^{p_n}$$ and $$\bar{z}^q=\bar{z}_1^{q_1}\bar{z}_2^{q_2}\dots\bar{z}_n^{q_n}$$, where $$p_s$$ and $$q_r$$ are nonnegative integers. Reviewer: Yuri I. Karlovich (Cuernavaca) The essential norm of multiplication operators acting on Orlicz sequence spaces https://zbmath.org/1485.47051 2022-06-24T15:10:38.853281Z "Ramos-Fernández, Julio C." https://zbmath.org/authors/?q=ai:ramos-fernandez.julio-c "Salas-Brown, Margot" https://zbmath.org/authors/?q=ai:salas-brown.margot Summary: We calculate the measure of non-compactness or the essential norm of the multiplication operator $M_u$ acting on Orlicz sequence spaces $$\ell^\varphi$$. As a consequence of our result, we obtain a known criteria for the compactness of multiplication operator acting on $$\ell^\varphi$$. Which weighted shifts are flat? https://zbmath.org/1485.47052 2022-06-24T15:10:38.853281Z "Shen, Hailong" https://zbmath.org/authors/?q=ai:shen.hailong "Li, Chunji" https://zbmath.org/authors/?q=ai:li.chunji Summary: The flatness property of unilateral weighted shifts is important to study the gaps between subnormality and hyponormality. In this paper, we first summerize the results on the flatness for some special kinds of a weighted shifts. And then, we consider the flatness property for a local-cubically hyponormal weighted shifts, which was introduced in [\textit{S. Baek} et al., Kyungpook Math. J. 59, No. 2, 315--324 (2019; Zbl 1441.47040)]. Let $$\alpha : {\sqrt{\frac{2}{3}}}, {\sqrt{\frac{2}{3}}}, \left\{{\sqrt{\frac{n+1}{n+2}}}\right\}^{\infty}_{n=2}$$ and let $$W_\alpha$$ be the associated weighted shift. We prove that $$W_\alpha$$ is a local-cubically hyponormal weighted shift $$W_\alpha$$ of order $$\theta={\frac{\pi}{4}}$$ by numerical calculation. A note on multiplication and composition operators in Orlicz spaces https://zbmath.org/1485.47053 2022-06-24T15:10:38.853281Z "Estaremi, Y." https://zbmath.org/authors/?q=ai:estaremi.yousef "Maghsoudi, S." https://zbmath.org/authors/?q=ai:maghsoudi.saeid "Rahmani, I." https://zbmath.org/authors/?q=ai:rahmani.iraj Summary: In this note, we give some results on the ascent and descent of multiplication and composition operators on Orlicz spaces. Some results of Lambert operators on $$L^p$$ spaces https://zbmath.org/1485.47054 2022-06-24T15:10:38.853281Z "Hosseini, Seyed Kamel" https://zbmath.org/authors/?q=ai:hosseini.seyed-kamel "Cheshmavar, Jahangir" https://zbmath.org/authors/?q=ai:cheshmavar.jahangir Summary: In this paper we provide necessary and sufficient conditions for Lambert operators to be invertible. Also, some properties of these type of operators will be investigated. Compact weighted Frobenius-Perron operators and their spectra https://zbmath.org/1485.47055 2022-06-24T15:10:38.853281Z "Jabbarzadeh, M. R." https://zbmath.org/authors/?q=ai:jabbarzadeh.mamed-rza-r|jabbarzadeh.mohammd-reza|jabbarzadeh.mohammad-reza "Emamalipour, H." https://zbmath.org/authors/?q=ai:emamalipour.hossein|emamalipour.hossain Summary: In this paper we characterize the compact weighted Frobenius-Perron operator $$P_\phi^u$$ on $$L_1(\Sigma)$$ and determine its spectra. Also, it is shown that every weakly compact weighted Frobenius-Perron operator on $$L_1(\Sigma)$$ is compact. $$L^p$$ compactness of Riesz transforms and their commutators related to generalized Schrödinger operators https://zbmath.org/1485.47056 2022-06-24T15:10:38.853281Z "Ding, Shanshan" https://zbmath.org/authors/?q=ai:ding.shanshan Summary: In this paper, the compactness of commutators $$[b, T]f(x) = b(x)Tf(x) - T(bf)(x)$$ on $${L^p}(\mathbb{R}^n)$$ for $$1 < p < \infty$$ is obtained, where $$T$$ is any of the Riesz transforms or their conjugates associated to the generalized Schrödinger operator $$-\Delta + \mu$$ with $$\mu$$ being a nonnegative Radon measure and $$b \in {\mathrm{VMO}}$$. The Riesz transform associated to the generalized Schrödinger operator is also compact on $${L^p}(\mathbb{H}^n)$$. $$\sigma$$-intertwinings, $$\sigma$$-cocycles and automatic continuity https://zbmath.org/1485.47057 2022-06-24T15:10:38.853281Z "Rad, Hussien Mahdavian" https://zbmath.org/authors/?q=ai:rad.hussien-mahdavian "Niknam, Assadollah" https://zbmath.org/authors/?q=ai:niknam.assadollah Summary: Let $$\mathcal{A}$$ be an algebra, $$\mathcal{X}$$ an $$\mathcal{A}$$-bimodule and $$\sigma: \mathcal{A} \to \mathcal{A}$$ a continuous homomorphism. In this paper, we show a continuous linear one to one correspondence between $$Z_{\sigma}^1 (\mathcal{A}, \mathcal{F})$$, the set of all module valued $$\sigma$$-derivations and $$LI_{\sigma}(\mathcal{A}, \mathcal{X})$$, the set of all left $$\sigma$$-intertwining mappings, where $$\mathcal{F}=B(\mathcal{A}_+, \mathcal{X})$$ and that $$B(\mathcal{A}_+, \mathcal{X})$$ is a $$\sigma(\mathcal{A})$$-bimodule. A~similar fact is proved between $$Z_{\sigma}^n (\mathcal{A}, \mathcal{F})$$, the set of all $$n$$-$$\sigma$$-cocycles, and $$LI_{\sigma}^n (\mathcal{A}, \mathcal{X})$$, the set of all $$\sigma$$-intertwining mappings in the last variables. Also there exists a linear homeomorphism between $$\mathfrak{Z}_{\sigma}^1 (\mathcal{A}, \mathcal{F})$$, the set of all continuous module valued $$\sigma$$-derivations, and $$B(\mathcal{A}, \mathcal{X})$$. Moreover, it is proved that the same relation satisfies between $$\mathfrak{Z}_{\sigma}^n (\mathcal{A}, \mathcal{F})$$ and $$B^n (\mathcal{A}, \mathcal{X})$$. Nonlinear $$\xi$$-Jordan $$*$$-triple derivable mappings on factor von Neumann algebras https://zbmath.org/1485.47058 2022-06-24T15:10:38.853281Z "Zhang, Fangjuan" https://zbmath.org/authors/?q=ai:zhang.fangjuan "Zhu, Xinhong" https://zbmath.org/authors/?q=ai:zhu.xinhong Summary: Let $$\mathcal{A}$$ be a factor von Neumann algebra and $$\xi$$ be a non-zero complex number. A~nonlinear map $$\phi:\mathcal{A} \to \mathcal{A}$$ has been demonstrated to satisfy $$\phi(A{\diamondsuit_\xi}B{\diamondsuit_\xi}C) = \phi(A){\diamondsuit_\xi}B{\diamondsuit_\xi}C + A{\diamondsuit_\xi} \phi(B){\diamondsuit_\xi}C + A{\diamondsuit_\xi}B{\diamondsuit_\xi}\phi(C)$$ for all $$A, B, C \in \mathcal{A}$$ if and only if $$\phi$$ is an additive $$*$$-derivation and $$\phi(\xi A) = \xi \phi(A)$$ for all $$A \in \mathcal{A}$$. Converting the properties in $$\mathcal{B}(\mathcal{H})$$ by operators on $$\mathcal{B}(\mathcal{H})$$ https://zbmath.org/1485.47059 2022-06-24T15:10:38.853281Z "Ansari-Piri, E." https://zbmath.org/authors/?q=ai:ansari-piri.esmaeil "Sanati, R. G." https://zbmath.org/authors/?q=ai:sanati.reza-ganjbakhs "Parsania, S." https://zbmath.org/authors/?q=ai:parsania.s Summary: The pair of operators on $$\mathcal{B}(\mathcal{H})$$ which are related to each other with respect to a specific property on $$\mathcal{B}(\mathcal{H})$$, have been studied before. In this paper, we study a pair of operators $$\varphi_1, \varphi_2$$ on $$\mathcal{B}(\mathcal{H})$$ which can convert some suitable properties to each other. For instance, we show that $$\varphi_1(T)$$ is a compact operator if and only if $$\varphi_2(T)$$ is compact, whenever $$\varphi_1(T)$$ is a Fredholm operator if and only if $$\varphi_2(T)$$ is a semi-Fredholm operator. Maps preserving fixed points of generalized product of operators https://zbmath.org/1485.47060 2022-06-24T15:10:38.853281Z "Bouramdane, Y." https://zbmath.org/authors/?q=ai:bouramdane.y "Ech-Cherif El Kettani, M." https://zbmath.org/authors/?q=ai:ech-cherif-elkettani.mustapha|el-kettani.m-ech-cherif "Elhiri, A." https://zbmath.org/authors/?q=ai:elhiri.a "Lahssaini, A." https://zbmath.org/authors/?q=ai:lahssaini.aziz Summary: Let $$\mathcal{B}(X)$$ be the algebra of all bounded linear operators in a complex Banach space $$X$$. For $$A\in\mathcal{B}(X)$$ let $$F(A)$$ be the subspace of fixed points of $$A$$. For an integer $$k\geq 2$$, let $$(i_1,\dots, i_m)$$ be a finite sequence with terms chosen from $$\{1,\dots,k\}$$ and assume at least one of the terms in $$(i_1,\dots,i_m)$$ appears exactly once. The generalized product of $$k$$ operators $$A_1,\dots, A_k\in\mathcal{B}(X)$$ is defined by $A_1* A_2*\cdots* A_k = A_{i_1} A_{i_2}\cdots A_{i_m}$ and includes the usual product and the triple product. We characterize the form of maps from $$\mathcal{B}(X)$$ onto itself satisfying $F (\varphi(A_1)* \cdots*\varphi(A_k)) = F (A_1*\cdots* A_k)$ for all $$A_1,\dots, A_k\in\mathcal{B}(X)$$. Nonlinear preservers of pseudospectral radius on Banach spaces https://zbmath.org/1485.47061 2022-06-24T15:10:38.853281Z "Costara, Constantin" https://zbmath.org/authors/?q=ai:costara.constantin Let $${\mathcal L}(X)$$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $$X$$. For $$\epsilon>0$$, the $$\epsilon$$--pseudospectrum of an operator $$T\in{\mathcal L}(X)$$ is $$\sigma_\epsilon(T)=\left\{\lambda\in\mathbb C:\|(\lambda-T)^{-1}\|\geq\frac{1}{\epsilon}\right\},$$ with the convention that $$\|(\lambda-T)^{-1}\|=\infty$$ if $$\lambda-T$$ is not invertible. This set is compact and the $$\epsilon$$-pseudospectral radius of $$T$$ is $$r_\epsilon(T):=\max\big\{|\lambda|:\lambda\in\sigma_\epsilon(T)\big\}.$$ The author proves that a surjective map $$\varphi$$ on $${\mathcal L}(X)$$ satisfies $r_\epsilon(\varphi(T)\varphi(S))=r_\epsilon(TS),\ T,S\in{\mathcal L}(X),$ if and only if there exist a complex-valued functional $$\xi$$ on $${\mathcal L}(X)$$ and a bijective linear or conjugate linear isometry $$U$$ on $$X$$ such that $$|\xi(T)|=1$$ and $$\varphi(T)=\xi(T)UTU^{-1}$$ for all $$T\in{\mathcal L}(X)$$. When $$X=\mathbb{C}^n$$ is a finite-dimensional space with any norm and $$n\geq3$$, he obtains a similar result but without the surjectivity assumption on $$\varphi$$. The proof of these results uses several auxiliary lemmas, some of them are new. Among the new ones, the author proves that an operator $$T\in{\mathcal L}(X)$$ is zero if and only if $$r_\epsilon(T)=\epsilon$$. Reviewer: Abdellatif Bourhim (Syracuse) On the pseudospectrum preservers https://zbmath.org/1485.47062 2022-06-24T15:10:38.853281Z "Ech-Chérif El Kettani, Mustapha" https://zbmath.org/authors/?q=ai:ech-cherif-elkettani.mustapha "Lahssaini, Aziz" https://zbmath.org/authors/?q=ai:lahssaini.aziz Summary: Let $$X$$ and $$Y$$ be two complex Banach spaces, and let $$B(X)$$ denotes the algebra of all bounded linear operators on $$X$$. We characterize additive maps from $$B(X)$$ onto $$B(Y)$$ compressing the pseudospectrum subsets $$\Delta_\epsilon(\cdot)$$, where $$\Delta_\epsilon (\cdot)$$ stands for any one of the spectral functions $$\sigma_\epsilon (\cdot)$$, $$\sigma^1_\epsilon(\cdot)$$ and $$\sigma^r_\epsilon (\cdot)$$ for some $$\epsilon >0$$. We also characterize the additive (resp., non-linear) maps from $$B(X)$$ onto $$B(Y)$$ preserving the pseudospectrum $$\sigma_\epsilon (\cdot)$$ of generalized products of operators for some $$\epsilon >0$$ (resp., for every $$\epsilon >0$$). Additive maps preserving zero skew $$\xi$$-Lie products https://zbmath.org/1485.47063 2022-06-24T15:10:38.853281Z "Qi, Xiaofei" https://zbmath.org/authors/?q=ai:qi.xiaofei "Hou, Jinchuan" https://zbmath.org/authors/?q=ai:hou.jinchuan "Cui, Jianlian" https://zbmath.org/authors/?q=ai:cui.jianlian Summary: Let $$H$$ and $$K$$ be complex Hilbert spaces with dimensions greater than 2, and $$\xi\in \mathbb{C}$$. Assume that $$\Phi:\mathcal{B}(H)\to \mathcal{B}(K)$$ is an additive surjective map satisfying that, for any $$A, B\in \mathcal{B}(H)$$, $$AB=\xi BA^*\Leftrightarrow \Phi(A)\Phi(B)=\xi \Phi(B)\Phi(A)^*$$. We show that (1) if $$\xi= 1$$, then there exist a unitary or an anti-unitary operator $$U: H\to K$$ and a nonzero real number $$c$$ such that $$\Phi(A)=c UAU^*$$ for all $$A\in \mathcal{B}(H)$$; (2) if $$\xi \in \mathbb{R} \backslash \{1\}$$ and $$\Phi$$ is unital, then there is a unitary or an anti-unitary operator $$U: H\to K$$ such that $$\Phi(A)=UAU^*$$ for all $$A\in \mathcal{B}(H)$$; (3) if $$\xi \in \mathbb{C} \backslash \mathbb{R}$$ and $$\Phi$$ is unital, then there is a unitary operator $$U: H\to K$$ such that $$\Phi(A)=UAU^*$$ for all $$A\in \mathcal{B}(H)$$. Asymptotics of eigenvalues of perturbed bilaplacian in the 1D lattice https://zbmath.org/1485.47064 2022-06-24T15:10:38.853281Z "Khalkhuzhaev, A. M." https://zbmath.org/authors/?q=ai:khalkhuzhaev.ahmad-m "Pardabaev, M. A." https://zbmath.org/authors/?q=ai:pardabaev.m-a Summary: We study eigenvalues of the Schrödinger-type operator $\widehat{\mathbf{h}}_\mu := \widehat{\Delta}\widehat{\Delta}-\mu\widehat{\mathbf{v}}_{ab},\; \mu > 0,$ in one dimensional lattice $$\mathbb{Z}$$, where $$\widehat{\Delta}$$ is the discrete Laplacian on $$\mathbb{Z}$$ and $$\widehat{\mathbf{v}}_{ab}$$ is a rank one-operator depending on nonzero real parameters $$a$$ and $$b$$. We prove that the number $\mu_0 := \begin{cases} 0,& a\ne b,\\ \frac1{b^2} ,& a = b,\end{cases}$ is the coupling constant threshold, i.e., for any $$\mu\in (0,\mu_0]$$ the discrete spectrum of $$\widehat{\mathbf{h}}_\mu$$ is empty and for any $$\mu >\mu_0$$ the discrete spectrum of $$\mathbf{h}_\mu$$ is a singleton $$e(\mu)$$, and $$e(\mu)<0$$ for $$\mu>\mu_0$$. Moreover, we study the properties of $$e(\mu)$$ as a function of $$\mu$$, in particular, we find the asymptotics of $$e(\mu)$$ as $$\mu\searrow \mu_0$$ and $$\mu\to+\infty$$. Invertibility of infinite-dimensional Hamiltonian operator and its applications https://zbmath.org/1485.47065 2022-06-24T15:10:38.853281Z "Wu, Deyu" https://zbmath.org/authors/?q=ai:wu.deyu "Alatancang" https://zbmath.org/authors/?q=ai:chen.alatancang Summary: In this paper the invertibility of infinite-dimensional Hamiltonian operators which arise in infinite-dimensional Hamiltonian systems are studied. The results illustrate the spectral properties of infinite dimensional Hamiltonian operators and characterize the compactness of inverse by applying its entries. In the end, applications to the invertibility of Dirac operators are given. Eventual domination for linear evolution equations https://zbmath.org/1485.47068 2022-06-24T15:10:38.853281Z "Glück, Jochen" https://zbmath.org/authors/?q=ai:gluck.jochen "Mugnolo, Delio" https://zbmath.org/authors/?q=ai:mugnolo.delio A Banach lattice is a lattice that is at the same time a Banach space with a norm satisfying $$|x| \le |y| \Rightarrow \|x\| \le \|y\|$$. The primary examples are $$L^p$$ spaces and spaces of continuous functions. If $$\{e^{tA}: t \ge 0\}$$ and $$\{e^{tB}: t \ge 0\}$$ are $$C_0$$ semigroups, then the second dominates the first if $|e^{tA}f| \le e^{tB}|f|. \tag{1}$ The semigroup $$\{e^{tA}: t \ge 0\}$$ is positive if $f \ge 0 \ \Rightarrow \ e^{tA}f \ge 0. \tag{2}$ If this inequality only holds for sufficiently large $$t$$, then the semigroup is eventually positive. Likewise, if (1) holds for sufficiently large $$t$$, then $$\{e^{tB}: t \ge 0\}$$ eventually dominates $$\{e^{tA}: t \ge 0\}$$. Finally, if (1) holds for $$t \ge t_0$$ independent of $$f$$, the semigroup's eventual positivity is uniform; likewise, if (2) holds for $$t \ge t_0$$ independent of $$f$$, then the eventual domination relation is uniform. As a motivation, the authors introduce two examples. One is the heat equation $u_t(t, x) = u_{xx}(t, x)$ in $$L^2(0, 1)$$ with two types of boundary condition: \begin{align*} \text{Dirichlet-Neumann condition:}\quad &u(t, 0) = 0, \ u_x(t, 1) = 0\;\; (t \ge 0) \tag{3} \\ \text{Periodic:}\quad &u(t, 0) = u(t, 1), \ u_x(t, 0) = u_x(t, 1)\;\;(t \ge 0) \tag{4} \end{align*} If $$A$$ (resp., $$B$$) is the Laplacian with conditions (3) (resp., with conditions (4)), then neither semigroup dominates the other; however, we have $$e^{tB}f \ge e^{tA}f$$ for $$f \ge 0$$, $$t \ge t_0$$. In the other example, $$\Omega$$ is an $$m$$-dimensional domain, $$B = \Delta$$ is the Dirichlet Laplacian and $$A = 2B$$. The results in this paper are on characterization of eventual domination in terms of assumptions involving the spectrum, smoothing properties, and eventual positivity of the operators $$A, B$$ involved. Reviewer: Hector O. Fattorini (Los Angeles) A dynamical approach to the Perron-Frobenius theory and generalized Krein-Rutman type theorems https://zbmath.org/1485.47124 2022-06-24T15:10:38.853281Z "Li, Desheng" https://zbmath.org/authors/?q=ai:li.desheng "Jia, Mo" https://zbmath.org/authors/?q=ai:jia.mo The authors develop a self-contained elementary dynamical approach towards the theory of the finite-dimensional complex version of the Perron-Frobenius theory. This enables them to prove a generalized finite-dimensional Krein-Rutman (KR) type theorem for complex operators, under rather general assumptions. In the presence of a rotational strong positivity assumption, this result immediately yields a refined complex version of the KR theorem. Next, extensions of these results to infinite-dimensional spaces are studied. Then the case of real bounded linear operators is considered. Employing the complexification technique, one obtains a generalized KR type theorem. Reviewer: K. C. Sivakumar (Chennai) Superunitary representations of Heisenberg supergroups https://zbmath.org/1485.81036 2022-06-24T15:10:38.853281Z "de Goursac, Axel" https://zbmath.org/authors/?q=ai:de-goursac.axel-marcillaud "Michel, Jean-Philippe" https://zbmath.org/authors/?q=ai:michel.jean-philippe Summary: Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone-von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs. Infinite-time admissibility of the Gurtin-Pipkin systems in Hilbert spaces https://zbmath.org/1485.93265 2022-06-24T15:10:38.853281Z "Chen, Jian-Hua" https://zbmath.org/authors/?q=ai:chen.jianhua "Fu, Lin" https://zbmath.org/authors/?q=ai:fu.lin "Zhou, Hua-Cheng" https://zbmath.org/authors/?q=ai:zhou.hua-cheng