Recent zbMATH articles in MSC 47Bhttps://zbmath.org/atom/cc/47B2023-11-13T18:48:18.785376ZWerkzeugFlat bands of periodic graphshttps://zbmath.org/1521.051072023-11-13T18:48:18.785376Z"Sabri, Mostafa"https://zbmath.org/authors/?q=ai:sabri.mostafa"Youssef, Pierre"https://zbmath.org/authors/?q=ai:youssef.pierreSummary: We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands and some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell, and we compute the list of such bands for small cells. We next discuss the stability and rarity of flat bands in special cases. Additional folklore results are proved, and many questions are still open.
{\copyright 2023 American Institute of Physics}Inequalities related to the \(\mathrm{S}\)-divergencehttps://zbmath.org/1521.150172023-11-13T18:48:18.785376Z"Feng, Lin"https://zbmath.org/authors/?q=ai:feng.lin"Lii, Lei"https://zbmath.org/authors/?q=ai:lii.lei"Zhang, Junming"https://zbmath.org/authors/?q=ai:zhang.junming"Zhang, Yuanming"https://zbmath.org/authors/?q=ai:zhang.yuanmingSummary: The \(\mathrm{S}\)-Divergence is a distance like function on the convex cone of positive definite matrices, which is motivated from convex optimization. In this paper, we will prove some inequalities for Kubo-Ando means with respect to the square root of the \(\mathrm{S}\)-Divergence.On norm inequalities related to the geometric meanhttps://zbmath.org/1521.150192023-11-13T18:48:18.785376Z"Freewan, Shaima'a"https://zbmath.org/authors/?q=ai:freewan.shaimaa"Hayajneh, Mostafa"https://zbmath.org/authors/?q=ai:hayajneh.mostafaThe authors give an affirmative answer to a conjecture posed by \textit{T. H. Dinh} et al. [Czech. Math. J. 66, No. 3, 777--792 (2016; Zbl 1413.15039)].
For this purpose, they prove a more general norm inequality related to the geometric mean, which leads to some known results as well.
Reviewer: Venus Kaleibary (Tehran)Preservers of the \(p\)-power and the Wasserstein means on \(2 \times 2\) matriceshttps://zbmath.org/1521.150242023-11-13T18:48:18.785376Z"Simon, Richárd"https://zbmath.org/authors/?q=ai:simon.richard-m"Virosztek, Dániel"https://zbmath.org/authors/?q=ai:virosztek.danielSummary: In one of his recent papers, \textit{L. Molnár} [Linear Algebra Appl. 430, No. 11--12, 3058--3065 (2009; Zbl 1182.47035)]
showed that if \(\mathcal{A}\) is a von Neumann algebra without \(I_1, I_2\)-type direct summands, then any function from the positive definite cone of \(\mathcal{A}\) to the positive real numbers preserving the Kubo-Ando power mean, for some \(0 \neq p \in (-1,1)\), is necessarily constant. It was shown in that paper that \(I_1\)-type algebras admit nontrivial \(p\)-power mean preserving functionals, and it was conjectured that \(I_2\)-type algebras admit only constant \(p\)-power mean preserving functionals. We confirm the latter. A similar result occurred in another recent paper of \textit{L. Molnár} [Aequationes Math. 94, No. 4, 703--722 (2020; Zbl 1518.47065)] concerning the Wasserstein mean. We prove the conjecture for \(I_2\)-type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in \(C^\ast\)-algebras.Distribution results for a special class of matrix sequences: joining approximation theory and asymptotic linear algebrahttps://zbmath.org/1521.150252023-11-13T18:48:18.785376Z"Schiavoni-Piazza, Alec Jacopo Almo"https://zbmath.org/authors/?q=ai:schiavoni-piazza.alec-jacopo-almo"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoSummary: In a recent paper, \textit{D. S. Lubinsky} [Linear Algebra Appl. 633, 332--365 (2022; Zbl 1478.15043)]
proved eigenvalue distribution results for a class of Hankel matrix sequences arising in several applications, ranging from Padé approximation to orthogonal polynomials and complex analysis. The results appeared in Linear Algebra and its Applications, and indeed many of the statements, whose origin belongs to the field of approximation theory and complex analysis, contain deep results in (asymptotic) linear algebra. Here we make an analysis of a part of these findings by combining his derivation with previous results in asymptotic linear algebra, showing that the use of an already available machinery shortens considerably the considered part of the derivations. Remarks and few additional results are also provided, in the spirit of bridging (numerical and asymptotic) linear algebra results and those coming from analysis and pure approximation theory.Eventual cone invariance revisitedhttps://zbmath.org/1521.150282023-11-13T18:48:18.785376Z"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochen"Hölz, Julian"https://zbmath.org/authors/?q=ai:holz.julianSummary: We consider finite-dimensional real vector spaces \(X\) ordered by a closed cone \(X_+\) with non-empty interior and study eventual nonnegativity of matrix semigroups \((e^{t A})_{t \geq 0}\) with respect to this cone.
Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that \(e^{t A}\) maps \(X_+\) into \(X_+\) for all sufficiently large times \(t\); (ii) individual eventual nonnegativity means that for each \(x \in X_+\) the vector \(e^{t A} x\) is in \(X_+\) for all \(t\) larger than an \(x\)-dependent time \(t_0\); and (iii) weak eventual nonnegativity means that for each \(x \in X_+\) and each functional \(x^\prime\) in the dual cone \(X_+^\prime\) the value \(\langle x^\prime, e^{t A} x \rangle\) is in \([0, \infty)\) for all \(t\) larger than an \(x\)- and \(x^\prime\)-dependent time \(t_0\). Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different.
Our second contribution is to show that typical Perron-Frobenius like properties remain valid under the weakest of the above notions.
Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that the uniform, individual and weak versions of this property are -- in contrast to the nonnegative case -- equivalent, and that they can be characterized by spectral properties.The capacity of quiver representations and Brascamp-Lieb constantshttps://zbmath.org/1521.160092023-11-13T18:48:18.785376Z"Chindris, Calin"https://zbmath.org/authors/?q=ai:chindris.calin"Derksen, Harm"https://zbmath.org/authors/?q=ai:derksen.harmGiven a real representation \(V\) of a bipartite quiver \(Q\), along with an integral weight \(\sigma\) of \(Q\) orthogonal to the dimension vector of \(V\), the authors introduce the Brascamp-Lieb (BL) operator \(T_{V,\sigma}\) associated to this datum, and study its capacity \(D_Q(V,\sigma)\). They characterise the positivity of \(D_Q(V,\sigma)\) by the \(\sigma\)-semi-stability of \(V\) (Theorem 1) and give a character formula in case the capacity is positive (Theorem 2). In the case of the \(m\)-subspace quiver, \(D_Q(V,\sigma)\) is related to the BL constants in the \(m\)-multilinear BL inequality in analysis.
The article is organised in five sections.
In the introduction, the authors motivate their studies by a result and comment in [\textit{J. Bennett} et al., Geom. Funct. Anal. 17, No. 5, 1343--1415 (2008; Zbl 1132.26006)], where the relevance of a ``deeper theory of representations'' to the Brascam-Lieb inequality in harmonic analysis and its theory is hinted at. Here, they also give a concise overview of notation and their results.
The second section defines the BL operator \(T_{V,\sigma}\) associated to real \(Q\)-representation \(V\) and weight \(\sigma\) (inspired by [\textit{A. Garg} et al., Geom. Funct. Anal. 28, No. 1, 100--145 (2018; Zbl 1387.68133)]) and its capacity (Definition 4). The section then proves Theorem 1, using a criterion also from [loc. cit.].
Section 3 provides an explicit formula for the capacity of a quiver datum \((V, \sigma)\) (Lemma 8, Corollary 9) and defines the notion of BL constants \(BL_Q\) for quiver datum (Definition 10).
In Section 4, the notion of a geometric quiver datum is introduced (Definition 12). In the language of quiver invariant theory, generalising a result of [\textit{A. D. King}, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)] and using a quiver version of a result of Kempf-Ness on closed orbits, the authors prove Theorem 2. This result includes a factorisation of the capacity of quiver data, the equivalence of existence of Gaussian extremals for \((V, \sigma)\) and \(V\) being \(\sigma\)-polystable, and that the uniqueness of Gaussian extremals implies that \(V\) is \(\sigma\)-stable.
Finally, in Section 5, the authors rephrase their main results in terms of BL constants and mention first applications [\textit{C. Chindris} and \textit{D. Kline}, J. Algebra 577, 210--236 (2021; Zbl 1467.16014)] and [\textit{C. Chindris} and \textit{D. Kline}, J. Pure Appl. Algebra 227, No. 3, Article ID 107234 (2023; Zbl 1517.16012)].
Reviewer: Sebastian Eckert (Bielefeld)Improved matrix inequalities using radical convexityhttps://zbmath.org/1521.260062023-11-13T18:48:18.785376Z"Sababheh, Mohammad"https://zbmath.org/authors/?q=ai:sababheh.mohammad-s"Furuichi, Shigeru"https://zbmath.org/authors/?q=ai:furuichi.shigeru"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-rezaThe authors present several inequalities for \(2\)-radical convex functions which refine some known inequalities for convex functions.
Let \(f:[0,\infty ) \rightarrow [0,\infty)\) be a continuous function with \(f(0)=0\) and let \(p\geq 1\) be a fixed number. If the function \(g(x)=f(x^{1/p})\) is convex on \([0,\infty)\), we say that \(f\) is \(p\)-radical convex.
Theorem 1. Let \(f\) be a 2-radical convex function.
(i) If \(a,b\geq 0\) and \(t\in [0,1]\), then \[ f((1-t)a+tb)+f\left( \sqrt{\frac{r|1-2t|}{2}}|a-b|\right) + 2r\left( \frac{f(a)+f(b)}{2} -f\left( \frac{a+b}{2}\right) \right) \leq (1-t)f(a) +t f(b), \] where \(r=\min \{ t,1-t\}\).
(ii) If \(a,b> 0\) and \(t\in [0,1]\), then \[ f((1-t)a+tb) \leq \frac{((1-t)a+tb)^2}{(1-t)a^2+tb^2} ((1-t)f(a) +tf(b)).\]
(iii) If (\(0<a<b\) and \(t>1\)) or (\(0<b<a\) and \(t<0\)), then for any \(t\not\in [0,1]\) \[ (1-t)f(a) +tf(b) \leq \frac{(1-t)a^2+tb^2}{((1-t)a+tb)^2} f((1-t)a+tb) .\] \\
Similar inequalities for matrices are also given. If \(X, Y \in \mathcal{M}_n\) are Hermitian matrices, then we say that \(\lambda (X) \leq \lambda(Y) \) if \(\lambda_j(X) \leq \lambda_j(Y)\) for each \(j=1,\ldots ,n\), where \(\lambda_j(X)\) is the \(j\)th largest eigenvalue of \(X\).
Theorem 2. Let \(f\) be a 2-radical convex function. Let \(A,B \in \mathcal{M}_n\) be positive definite matrices, \(m,m',M,M'\in \mathbb{R}\) and \(t\in [0,1]\).
(i) If \(0<m'I \leq A \leq mI \leq MI\leq B\leq M'I\), then \[ \lambda(f((1-t)A+tB))\leq \frac{((1-t)m+tM)^2}{(1-t)m^2+tM^2} \lambda(((1-t)f(A)+tf(B))). \]
(ii) If \(0<m'I \leq B \leq mI \leq MI\leq A\leq M'I\), then \[ \lambda(f((1-t)A+tB))\leq \frac{((1-t)M+tm)^2}{(1-t)M^2+tm^2} \lambda(((1-t)f(A)+tf(B))). \]
At the end of the article, some examples are given to illustrate the main result.
Reviewer: Sanja Varošanec (Zagreb)Schur analysis and discrete analytic functions: rational functions and co-isometric realizationshttps://zbmath.org/1521.300572023-11-13T18:48:18.785376Z"Alpay, Daniel"https://zbmath.org/authors/?q=ai:alpay.daniel"Volok, Dan"https://zbmath.org/authors/?q=ai:volok.danSummary: We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers.Weighted composition operators from Dirichlet-Zygmund-type spaces into Stević-type spaceshttps://zbmath.org/1521.300692023-11-13T18:48:18.785376Z"Zhu, Xiangling"https://zbmath.org/authors/?q=ai:zhu.xianglingSummary: A family of Zygmund-type spaces, called Dirichlet-Zygmund-type spaces, are introduced. The boundedness, compactness and the essential norm of weighted composition operators from Dirichlet-Zygmund-type spaces into Stević-type spaces are also investigated in this paper.On the spectrum of the differential operators of even order with periodic matrix coefficientshttps://zbmath.org/1521.340782023-11-13T18:48:18.785376Z"Veliev, O. A."https://zbmath.org/authors/?q=ai:veliev.oktay-alishThe paper deals with the differential operator \(L\) generated in the space \(L_2^m(\mathbb{R})\) of vector-valued functions by the formally self-adjoint differential expression \[(-i)^{2\nu}y^{(2\nu)}(x)+\displaystyle\sum_{k=2}^{2\nu}P_k(x)y^{(2\nu-k)}(x),\] where \(\nu>1\) and \(P_k(x)\) is a \(m\times m\) matrix with summable entries \(p_{k,i,j}\) which satisfy the periodicity conditions \(p_{k,i,j}(x+1)=p_{k,i,j}(x)\) for all \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,m\), for \(k=2,3,\ldots,2\nu\). The author investigates the band functions, Bloch functions and the spectrum of operator \(L\).
Reviewer: Rodica Luca (Iaşi)Invariant measures for uncountable random interval homeomorphismshttps://zbmath.org/1521.370522023-11-13T18:48:18.785376Z"Morawiec, Janusz"https://zbmath.org/authors/?q=ai:morawiec.janusz"Szarek, Tomasz"https://zbmath.org/authors/?q=ai:szarek.tomasz-jakub|szarek.tomasz-zacharySummary: A necessary and sufficient condition for the iterated function system \(\{f(\cdot, \omega)|\omega\in\Omega\}\) with probability \(P\) to have exactly one invariant measure \(\mu_\ast\) with \(\mu_\ast((0,1)) = 1\) is given. The main novelty lies in the fact that we only require the transformations \(f(\cdot, \omega)\) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of \(\Omega\). In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.Estimate for some integral operators and their commutators on generalized fractional mixed Morrey spaceshttps://zbmath.org/1521.420152023-11-13T18:48:18.785376Z"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghui"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangpingIn this paper, the authors establish the definition of a generalized fractional mixed Morrey space and show that the Calderón-Zygmund integral operators, the fractional integral operator, and their commutators associated with BMO functions are bounded on such space.
Finally, the boundedness of the fractional maximal operator and its commutator associated with a BMO function and of the Calderón-Zygmund operator commutator associated with Lipschitz functions on generalized fractional mixed Morrey space is also established.
Reviewer: Qingze Lin (Guangzhou)A new vanishing Lipschitz-type subspace of BMO and compactness of bilinear commutatorshttps://zbmath.org/1521.420212023-11-13T18:48:18.785376Z"Tao, Jin"https://zbmath.org/authors/?q=ai:tao.jin"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongThe authors introduce a new vanishing Lipschitz-type space and obtain its characterization. As an application, the authors obtain the compactness of commutators of bilinear Calderón-Zygmund operators with functions in the new vanishing Lipschitz-type space.
Reviewer: Jingshi Xu (Guilin)Vector-valued numerical radius and \(\sigma\)-porosityhttps://zbmath.org/1521.460032023-11-13T18:48:18.785376Z"Bachir, Mohammed"https://zbmath.org/authors/?q=ai:bachir.mohammedThe author proves that, assuming that the Banach space \(X\) is a uniformly convex and uniformly smooth space, the set of all bounded linear operators attaining their numerical radius is not only dense (which is already known due to \textit{C.~S. Cardassi} [Bull. Aust. Math. Soc. 31, 1--3 (1985; Zbl 0557.47003)] or, later, \textit{S.~K. Kim} et al. [Abstr. Appl. Anal. 2014, Article ID 479208, 15~p. (2014; Zbl 1473.46011)]), but also it is the complement of a \(\sigma\)-porous subset. Moreover, the author introduces the notion of numerical radius to a large class of vector-valued operators defined from \(X \times X^*\) into a Banach space \(W\) and also provides similar results in this new direction. The Bishop-Phelps-Bollobás property for the numerical radius is also considered.
Reviewer: Sheldon Dantas (València)A generalized ACK structure and the denseness of norm attaining operatorshttps://zbmath.org/1521.460042023-11-13T18:48:18.785376Z"Choi, Geunsu"https://zbmath.org/authors/?q=ai:choi.geunsu"Jung, Mingu"https://zbmath.org/authors/?q=ai:jung.minguThe authors introduce the so-called quasi-ACK structure, which is a generalization of a concept introduced not long ago by
\textit{B.~Cascales} et al. [J. Funct. Anal. 274, No.~3, 863--888 (2018; Zbl 1396.46006)].
After providing basic and essential results for this new concept, the authors establish new examples concerning the density of norm attaining operators. They also provide new examples of Banach spaces satisfying the Lindenstrauss property B for compact operators as well as universal BPB range spaces for certain ideals after considering the problem for vector-valued holomorphic functions.
Reviewer: Sheldon Dantas (València)Nuclear operators on the Banach space of vector-valued essentially bounded measurable functionshttps://zbmath.org/1521.460192023-11-13T18:48:18.785376Z"Nowak, Marian"https://zbmath.org/authors/?q=ai:nowak.marianSummary: Let \((\Omega,\Sigma,\mu)\) be a finite measure space and \(\mathcal{N}(X,Y)\) stand for the Banach space of all nuclear operators between Banach spaces \(X\) and \(Y\), equipped with the nuclear norm \(\|\cdot \|_{nuc}\). It is shown that if \(X^*\) has the Radon-Nikodym property, then a bounded linear operator \(T:L^\infty (\mu,X)\rightarrow Y\) is a \(\sigma\)-order continuous nuclear operator between the Banach spaces \(L^\infty(\mu,X)\) and \(Y\) if and only if its representing measure \(m_T\) has a Radon-Nikodym derivative in \(L^1(\mu,\mathcal{N}(X,Y))\) and if and only if \(T\) has a Bochner kernel in \(L^1(\mu,\mathcal{N}(X,Y))\).On the maximal numerical range of the bimultiplication \(M_{2,A,B}\)https://zbmath.org/1521.470052023-11-13T18:48:18.785376Z"Baghdad, Abderrahim"https://zbmath.org/authors/?q=ai:baghdad.abderrahim"Chraibi Kaadoud, Mohamed"https://zbmath.org/authors/?q=ai:chraibi-kaadoud.mohamedSummary: Let \(\mathcal{B}(\mathcal{H})\) denote the algebra of all bounded linear operators acting on a complex Hilbert space \(\mathcal{H}\). For \(A,B \in \mathcal{B}(\mathcal{H})\), define the bimultiplication operator \(M_{2,A,B}\) on the class of Hilbert-Schmidt operators by \(M_{2,A,B}(X)=AXB\). In this paper, we show that, if \(B\) is normal, then
\[
co(W_0 (A)W_0 (B)) \subseteq W_0 (M_{2,A,B}),
\]
where \(co\) stands for the convex hull and \(W_0 (.)\) denotes the maximal numerical range. If in addition, \(A\) is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation \(\delta_{2,A,B}\) on the class of Hilbert-Schmidt operators are also given.Numerical radius inequalities for operator matriceshttps://zbmath.org/1521.470082023-11-13T18:48:18.785376Z"Huang, Hong"https://zbmath.org/authors/?q=ai:huang.hong.1|huang.hong"Zhu, Zhi-Feng"https://zbmath.org/authors/?q=ai:zhu.zhifeng"Xu, Guo-Jin"https://zbmath.org/authors/?q=ai:xu.guojinSummary: In this paper, we firstly establish new numerical radius inequalities which refine a result of \textit{F. Kittaneh} in [Stud. Math. 168, No. 1, 73--80 (2005; Zbl 1072.47004)], then present some numerical radius inequalities involving non-negative increasing convex functions for \(n\times n\) operator matrices, which generalize the related results of \textit{K. Shebrawi} in [Linear Algebra Appl. 523, 1--12 (2017; Zbl 1453.47001)].Numerical radius parallelism of Hilbert space operatorshttps://zbmath.org/1521.470102023-11-13T18:48:18.785376Z"Mehrazin, Marzieh"https://zbmath.org/authors/?q=ai:mehrazin.marzieh"Amyari, Maryam"https://zbmath.org/authors/?q=ai:amyari.maryam"Zamani, Ali"https://zbmath.org/authors/?q=ai:zamani.ali|zamani.ali-rezaSummary: In this paper, we study the numerical radius parallelism for bounded linear operators on a Hilbert space \(\big (\mathscr{H}, \langle \cdot ,\cdot \rangle \big )\). More precisely, we consider bounded linear operators \(T\) and \(S\) which satisfy \(\omega (T + \lambda S) = \omega (T)+\omega (S)\) for some complex unit \(\lambda \), and is denoted by \(T \parallel_{\omega} S\). We show that \(T \parallel_{\omega} S\) if and only if there exists a sequence of unit vectors \(\{x_n\}\) in \(\mathscr{H}\) such that
\[
\lim_{n\rightarrow \infty}\big |\langle Tx_n, x_n\rangle \langle Sx_n, x_n\rangle \big | = \omega (T)\omega (S).
\]
We then apply it to give some applications.Numerical ranges of sum of two weighted composition operators on the Hardy space \(H^2\)https://zbmath.org/1521.470142023-11-13T18:48:18.785376Z"Shaabani, Mahmood Haji"https://zbmath.org/authors/?q=ai:shaabani.mahmood-haji"Vafaei, Narjes"https://zbmath.org/authors/?q=ai:vafaei.narjesSummary: Let \(\varphi\) be an analytic self-map of the open unit disk \(\mathbb{D}\) and let \(\psi\) be an analytic function on \(\mathbb{D}\). The weighted composition operator \(C_{\psi, \varphi}\) is the operator on the Hardy space \(H^2\) given by \(C_{\psi, \varphi} f = \psi f\circ\varphi\). Under some conditions on \(\varphi_1\) and \(\varphi_2\), we try tofind a subset of the numerical range of \(C_{\psi_1, \varphi_1} + C_{\psi_2, \varphi_2}\) and determine when zero lies in the interior of the numerical range of \(C_{\psi_1, \varphi_1} + C_{\psi_2, \varphi_2}\).\(A\)-numerical radius and product of semi-Hilbertian operatorshttps://zbmath.org/1521.470152023-11-13T18:48:18.785376Z"Zamani, Ali"https://zbmath.org/authors/?q=ai:zamani.ali|zamani.ali-rezaSummary: Let \(A\) be a positive bounded operator on a Hilbert space \(\big (\mathcal{H}, \langle \cdot, \cdot \rangle \big )\). The semi-inner product \({\langle x, y\rangle}_A := \langle Ax, y\rangle, x, y\in\mathcal{H}\), induces a seminorm \({\Vert \cdot \Vert }_A\) on \(\mathcal{H} \). Let \(w_A(T)\) denote the \(A\)-numerical radius of an operator \(T\) in the semi-Hilbertian space \(\big (\mathcal{H}, {\Vert \cdot \Vert}_A\big )\). In this paper, for any semi-Hilbertian operators \(T\) and \(S\), we show that \(w_A(TR) = w_A(SR)\) for all \((A\)-rank one) semi-Hilbertian operators \(R\) if and only if \(A^{1/2}T = \lambda A^{1/2}S\) for some complex unit \(\lambda \). From this result, we derive a number of consequences.Reducibility of WCE operators on \(L^2({\mathcal{F}})\)https://zbmath.org/1521.470172023-11-13T18:48:18.785376Z"Estaremi, Y."https://zbmath.org/authors/?q=ai:estaremi.yousefSummary: In this paper, we characterize the closed subspaces of \(L^2({\mathcal{F}})\) that reduce the operators of the form \(E^{{\mathcal{A}}}M_u\) in which \({\mathcal{A}}\) is a \(\sigma\)-subalgebra of \({\mathcal{F}}\). We show that \(L^2(A)\) reduces \(E^{{\mathcal{A}}}M_u\) if and only if \(E^{{\mathcal{A}}}(\chi _A)=\chi _A\) on the support of \(E^{{\mathcal{A}}}(|u|^2)\), where \(A\in{\mathcal{F}}\). Also, some necessary and sufficient conditions are provided for \(L^2({\mathcal{B}})\) to reduce \(E^{{\mathcal{A}}}M_u\) for the \(\sigma\)-subalgebra \({\mathcal{B}}\) of \({\mathcal{F}}\) .Eigenvalue inequalities for \(n\)-tuple of matriceshttps://zbmath.org/1521.470212023-11-13T18:48:18.785376Z"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-reza"Sababheh, Mohammad"https://zbmath.org/authors/?q=ai:sababheh.mohammad-sSummary: The main goal of this article is to present several inequalities for the eigenvalues of several operators, when filtered through positive linear mappings, convex functions, superquadratic functions and asynchronous functions.Unbounded asymptotic equivalences of operator nets with applicationshttps://zbmath.org/1521.470222023-11-13T18:48:18.785376Z"Erkurşun-Özcan, Nazife"https://zbmath.org/authors/?q=ai:erkursun-ozcan.nazife"Gezer, Niyazi Anıl"https://zbmath.org/authors/?q=ai:gezer.niyazi-anilSummary: The present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences \(\mathfrak{c}\) and \(\mathfrak{d}\) on a vector lattice, we study \(\mathfrak{d}\)-asymptotic properties of operator nets formed by \(\mathfrak{c}\)-continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on \(\mathfrak{d}\)-martingale and \(\mathfrak{d}\)-Lotz-Räbiger nets.On non-selfadjoint operators with finite discrete spectrumhttps://zbmath.org/1521.470252023-11-13T18:48:18.785376Z"Bourget, Olivier"https://zbmath.org/authors/?q=ai:bourget.olivier"Sambou, Diomba"https://zbmath.org/authors/?q=ai:sambou.diomba"Taarabt, Amal"https://zbmath.org/authors/?q=ai:taarabt.amalSummary: We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrödinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable regularity conditions.
For the entire collection see [Zbl 1461.35006].On some extensions of the A-modelhttps://zbmath.org/1521.470282023-11-13T18:48:18.785376Z"Juršėnas, Rytis"https://zbmath.org/authors/?q=ai:jursenas.rytisGiven a lower semibounded selfadjoint operator \(L\) in a Hilbert space \({\mathfrak H}_0\), let \({\mathfrak H}_{n+1}\subseteq{\mathfrak H}_n\) (\(n\in\mathbb{Z}\)) be the scale of Hilbert spaces associated with \(L\), where \(L\) has domain \(\mathrm{dom}L={\mathfrak H}_2\), and let \(\{\varphi_\sigma\}\) be a family of linearly independent functionals \(\varphi_\sigma \in{\mathfrak H}_{-m-2}\setminus{\mathfrak H}_{-m-1}\) (\(m\in\mathbb{N}\)), where \(\sigma\) ranges over the index set \({\mathcal S}\) of dimension \(d\in\mathbb{N}\). Then the symmetric restriction \(L_{\min}\subseteq L\) to the domain of \(f\in{\mathfrak H}_{m+2}\), such that \(\langle\varphi_\sigma,f\rangle=0\) for all \(\sigma\), is an essentially selfadjoint operator on \({\mathfrak H}_0\).
The paper deals with describing nontrivial extensions of \(L_{\min}\) (perturbations of \(L\)) in \({\mathfrak H}_0\). The cascade A-model for rank-\(d\) higher order singular perturbations is studied on the basis of ordinary boundary triples. Assuming the existence of orthogonal decompositions \({\mathfrak H}_n^+\oplus{\mathfrak H}_n^-\) of the Hilbert spaces \({\mathfrak H}_n\) (\(n\in\mathbb{N}\)) with orthogonal projections \(P_n^\pm:{\mathfrak H}_n\to{\mathfrak H}_n^\pm\) such that \(P_{n+1}^\pm\subseteq P_n^\pm\), nontrivial extensions in the \(A\)-model are constructed for the symmetric restrictions of \(L_{\min}\) in the subspaces. Nontrivial realizations of \(L_{\min}\) in the spaces \({\mathfrak H}_m^\pm\) are obtained with the use of the Krein-Naimark resolvent formula, the Weyl functions, the Krein \(Q\)-functions, and the generalized Nevanlinna functions.
Reviewer: Yuri I. Karlovich (Cuernavaca)Best approximation by diagonal operators in Schatten idealshttps://zbmath.org/1521.470292023-11-13T18:48:18.785376Z"Bottazzi, Tamara"https://zbmath.org/authors/?q=ai:bottazzi.tamaraAuthor's abstract: If \(\mathcal{X}\) is the set of compact or \(p\)-Schatten operators over a complex Hilbert separable space \(\mathcal{H}\), we study the existence and characterization properties of Hermitian \(A\in \mathcal{X}\) such that \[|||A|||\leq |||A + D|||, \textup{ for all } D \in \mathcal{D(X)},\] or, equivalently \[|||A|||= \min_{D\in \mathcal{D(X)}} |||A + D||| = \operatorname{dist}(A,\mathcal{D(X)}) ,\] where \(\mathcal{D(X)}\) is the subspace of diagonal operators of \(\mathcal{X}\) in any prefixed basis of \(\mathcal{H}\) and \(|||\cdot|||\) is the usual operator norm in each \(\mathcal{X}\). We use Birkhoff-James orthogonality as a tool to characterize and develop properties of these operators in each context. We also provide several illustrative examples.
Reviewer: Sen Zhu (Changchun)Wg-Drazin-star operator and its dualhttps://zbmath.org/1521.470312023-11-13T18:48:18.785376Z"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Zhang, Daochang"https://zbmath.org/authors/?q=ai:zhang.daochang"Hu, Jianping"https://zbmath.org/authors/?q=ai:hu.jianpingSummary: Our goal is to define new classes of bounded linear operators between two Hilbert spaces, solving corresponding systems of equations. Precisely, we introduce the Wg-Drazin-star operator and its dual, extending the notions of the \(W\)-weighted Drazin-star matrix and its dual for a rectangular matrix. We prove many characterizations and operator matrix representations of the Wg-Drazin-star operator and its dual. As special cases of the Wg-Drazin-star operator and its dual, the g-Drazin-star operator and its dual are presented and studied. Applying our new classes of operators, we also solve adequate equations. Thus, we generalize some well-known results and present new results.Regular covariant representations and their Wold-type decompositionhttps://zbmath.org/1521.470362023-11-13T18:48:18.785376Z"Rohilla, Azad"https://zbmath.org/authors/?q=ai:rohilla.azad"Veerabathiran, Shankar"https://zbmath.org/authors/?q=ai:veerabathiran.shankar"Trivedi, Harsh"https://zbmath.org/authors/?q=ai:trivedi.harshSummary: \textit{A. Olofsson} [Integral Equations Oper. Theory 51, No. 3, 395--409 (2005; Zbl 1079.47015)] introduced a growth condition regarding elements of an orbit for an expansive operator and generalized Richter's wandering subspace theorem. Later on, using the Moore-Penrose inverse, Ezzahraoui, Mbekhta, and Zerouali [\textit{H. Ezzahraoui} et al., J. Math. Anal. Appl. 430, No. 1, 483--499 (2015; Zbl 1329.47001)] extended the growth condition and obtained a Shimorin-Wold-type decomposition. Shimorin-Wold-type decomposition for completely bounded covariant representations, which are close to isometric representations, was obtained in [\textit{H. Trivedi} and \textit{S. Veerabathiran}, Integral Equations Oper. Theory 91, No. 4, Paper No. 35, 21 p. (2019; Zbl 1437.46057)]. This paper extends this decomposition for regular, completely bounded covariant representation having reduced minimum modulus \(\geq 1\) that satisfies the growth condition. To prove the decomposition, we introduce the terms regular, algebraic core, and reduced minimum modulus in the completely bounded covariant representation setting and work out several fundamental results. Consequently, we shall analyze the weighted unilateral shift introduced by \textit{P. S. Muhly} and \textit{B. Solel} [Integral Equations Oper. Theory 84, No. 4, 501--553 (2016; Zbl 1356.46042)] and introduce and explore a non-commutative weighted bilateral shift.Quasi-compactness of linear operators on Banach spaces: new properties and application to Markov chainshttps://zbmath.org/1521.470372023-11-13T18:48:18.785376Z"Mebarki, Leila"https://zbmath.org/authors/?q=ai:mebarki.leila"Messirdi, Bekkai"https://zbmath.org/authors/?q=ai:messirdi.bekkai"Benharrat, Mohammed"https://zbmath.org/authors/?q=ai:benharrat.mohammedThe paper contains a number of easy to prove statements on quasi compact operators on Banach spaces, including equivalence of different definitions and corollaries of known results. A~typical one: every polynomially compact operator is quasi compact. The authors provide some examples and implications for Markov chains. So the paper can be interesting for beginners as an introduction to the subject.
Reviewer: Mikhail M. Popov (Slupsk)Operator ideals generated by strongly Lorentz sequence spaceshttps://zbmath.org/1521.470382023-11-13T18:48:18.785376Z"Achour, Dahmane"https://zbmath.org/authors/?q=ai:achour.dahmane"Attallah, Aldjia"https://zbmath.org/authors/?q=ai:attallah.aldjiaThe authors present a new approach to the summability of operators. The ideal of strongly Lorentz summing operators between Banach spaces is introduced to study the adjoints of the Lorentz summing linear operators. Connections with the theory of absolutely summing operators are given.
Reviewer: Elhadj Dahia (Bou Saâda)\(p\)-Schatten commutators of projectionshttps://zbmath.org/1521.470392023-11-13T18:48:18.785376Z"Andruchow, Esteban"https://zbmath.org/authors/?q=ai:andruchow.esteban"Di Iorio y Lucero, María Eugenia"https://zbmath.org/authors/?q=ai:di-iorio-y-lucero.maria-eugeniaSummary: Let \(\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-\) be a fixed orthogonal decomposition of the complex separable Hilbert space \(\mathcal{H}\) in two infinite-dimensional subspaces. We study the geometry of the set \(\mathcal{P}^p\) of selfadjoint projections in the Banach algebra
\[
\mathcal{A}^p=\{A\in\mathcal{B}(\mathcal{H}): [A,E_+]\in\mathcal{B}_p(\mathcal{H})\},
\]
where \(E_+\) is the projection onto \(\mathcal{H}_+\) and \(\mathcal{B}_p(\mathcal{H})\) is the Schatten ideal of \(p\)-summable operators \((1\leq p<\infty)\). The norm in \(\mathcal{A}^p\) is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space \(\mathcal{P}^p\) is shown to be a differentiable \(C^\infty\) submanifold of \(\mathcal{A}^p\), and a homogeneous space of the group of unitary operators in \(\mathcal{A}^p\). The connected components of \(\mathcal{P}^p\) are characterized, by means of a partition of \(\mathcal{P}^p\) in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.A direct approach to positive normalised traces on simply generated idealshttps://zbmath.org/1521.470402023-11-13T18:48:18.785376Z"Usachev, Alexandr"https://zbmath.org/authors/?q=ai:usachev.aleksandrIn the present paper, the author develops a direct approach to all positive normalised traces on a larger class of principal ideals. Also, he discusses the optimal conditions for the existence of traces on these principal ideals. The paper offers a simple proof asserting that there are positive normalised traces which are not Dixmier traces and answers a question raised in [\textit{S. Lord} et al., in: Advances in noncommutative geometry. Based on the noncommutative geometry conference, Shanghai, China, March 23 -- April 7, 2017. On the occasion of Alain Connes' 70th Birthday. Cham: Springer. 491--583 (2019; Zbl 1457.46064)].
Reviewer: Elhadj Dahia (Bou Saâda)Factorization of generalized holomorphic curve and homogeneity of operatorshttps://zbmath.org/1521.470412023-11-13T18:48:18.785376Z"Hou, Yingli"https://zbmath.org/authors/?q=ai:hou.yingli"Ji, Kui"https://zbmath.org/authors/?q=ai:ji.kui"Zhao, Linlin"https://zbmath.org/authors/?q=ai:zhao.linlinUsing the tensor structure of holomorphic bundles, the paper gives a new characterization of homogeneous operators in the so-called Cowen-Douglas class \(B_n(\Omega)\). The paper also contains results about similarity of operators with Fredholm index \(n\) associated with Hermitian holomorphic bundles.
Reviewer: Kehe Zhu (Albany)Non \(C\)-normal operators are densehttps://zbmath.org/1521.470422023-11-13T18:48:18.785376Z"Amara, Zouheir"https://zbmath.org/authors/?q=ai:amara.zouheir"Oudghiri, Mourad"https://zbmath.org/authors/?q=ai:oudghiri.mouradIn this paper, the authors prove that, for every bounded linear operator \(A\) acting on a separable complex Hilbert space \(H\) (which is \(C\)-normal for some conjugation \(C\)) and, for every \(\varepsilon>0\), there exists a finite rank operator \(F\) with rank less or equal to 4 and \(\Vert F\Vert\leq \varepsilon\) such that \(A+F\) is in the class of operators that are \(C\)-normal with respect to some conjugation \(C\) and, as a consequence, \(A+F\) is neither complex symmetric nor skew-symmetric. The arguments of their proof are based on a refined polar decomposition for \(C\)-normal operators. Moreover, they prove that the set of operators that are not \(C\)-normal for any conjugation \(C\) on \(H\) is dense in the linear space of bounded linear operators acting on \(H\).
Reviewer: Bilel Krichen (Sfax)Pencils of pairs of projectionshttps://zbmath.org/1521.470432023-11-13T18:48:18.785376Z"Cui, Miaomiao"https://zbmath.org/authors/?q=ai:cui.miaomiao"Ji, Guoxing"https://zbmath.org/authors/?q=ai:ji.guoxingSummary: Let \(T\) be a self-adjoint operator on a complex Hilbert space \(\mathcal{H}\). In this paper, a sufficient and necessary condition for \(T\) to be (the value of) the pencil \(\lambda P+Q\) of a pair \(( P, Q)\) of projections at some point \(\lambda \in \mathbb{R}\backslash \{-1, 0\}\) is introduced. Then we give a representation of all pairs \((P, Q)\) of projections such that \(T=\lambda P+Q\) for a fixed real number \(\lambda \), and find that all such pairs constitute a connected set if \(\lambda \in \mathbb{R}\backslash \{-1, 0, 1\}\). Further, the von Neumann algebra generated by such pairs \((P,Q)\) is characterized. Moreover, we prove that there are at most two non-zero real numbers such that \(T\) is the pencil of a pair of projections at these numbers. Finally, we determine when there is only one such number.Essential norms of weighted composition operators from analytic function spaces into iterated weighted-type Banach spaceshttps://zbmath.org/1521.470442023-11-13T18:48:18.785376Z"Alyusof, Shams"https://zbmath.org/authors/?q=ai:alyusof.shams"Colonna, Flavia"https://zbmath.org/authors/?q=ai:colonna.flaviaSummary: In this work, we characterize the bounded and the compact weighted composition operators from a large class of Banach spaces \(X\) of analytic functions on the open unit disk \(\mathbb{D}\) into the weighted-type Banach spaces \(\mathcal{V}_{n, \mu}\), for \(n \geq 3\), where given a positive continuous function \(\mu\) on \(\mathbb{D}\), the sequence \(\{\mathcal{V}_{n, \mu}\}_{n \geq 0}\) is defined iteratively by \(f\in\mathcal{V}_{0, \mu}\) if and only if
\[
\|f\|_{\mathcal{V}_{0, \mu}} := \sup_{|z| < 1}\mu(z)|f(z)| < \infty,
\]
and for \(n \geq 1\), \(f\in\mathcal{V}_{n, \mu}\) if and only if \(f^\prime\in\mathcal{V}_{n-1, \mu}\), with norm \(\|f\|_{\mathcal{V}_{n, \mu}} := |f(0)| + \|f^\prime\|_{\mathcal{V}_{n-1, \mu}}\), thereby extending known results for the cases \(n = 0, 1, 2\). Under more restrictive conditions, we provide an approximation of the essential norm. We apply our results to the cases when \(X\) is a Hardy space and a weighted Bergman space.Some remarks on substitution and composition operatorshttps://zbmath.org/1521.470452023-11-13T18:48:18.785376Z"Appell, Jürgen"https://zbmath.org/authors/?q=ai:appell.jurgen-m"Brito, Belén López"https://zbmath.org/authors/?q=ai:brito.belen-lopez"Reinwand, Simon"https://zbmath.org/authors/?q=ai:reinwand.simon"Schöller, Kilian"https://zbmath.org/authors/?q=ai:scholler.kilianIn this paper, the authors study the linear substitution operator and nonlinear composition operator, namely,
\[
S_\varphi(f)=f\circ\varphi,\; \varphi:[0,1]\rightarrow [0,1],
\]
and
\[
C_g(f)=g\circ f,\; g:\mathbb{R}\rightarrow \mathbb{R},
\]
respectively. They show that these operators have a very different behavior in the space of continuous functions, Lipschitz functions, functions of bounded variation, and Baire class one functions. Moreover, they give examples and counterexamples which illustrate this behavior.
Reviewer: Mohamed Abdalla Darwish (Damanhour)Corrigendum to: ``Some remarks on substitution and composition operators''https://zbmath.org/1521.470462023-11-13T18:48:18.785376Z"Appell, Jürgen"https://zbmath.org/authors/?q=ai:appell.jurgen-m"Brito, Belén López"https://zbmath.org/authors/?q=ai:brito.belen-lopez"Reinwand, Simon"https://zbmath.org/authors/?q=ai:reinwand.simon"Schöller, Kilian"https://zbmath.org/authors/?q=ai:scholler.kilianThe authors correct an error in Proposition 3.3 (Part (c)) in their paper [ibid. 53, Paper No. 6, 25 p. (2021; Zbl 1521.47045)].
Reviewer: Mohamed Abdalla Darwish (Damanhour)Mean ergodic composition operators on spaces of holomorphic functions on a Banach spacehttps://zbmath.org/1521.470472023-11-13T18:48:18.785376Z"Jornet, David"https://zbmath.org/authors/?q=ai:jornet.david"Santacreu, Daniel"https://zbmath.org/authors/?q=ai:santacreu.daniel"Sevilla-Peris, Pablo"https://zbmath.org/authors/?q=ai:sevilla-peris.pabloFor a given complex Banach space \(X\), let \(B_X\) denote the open unit ball of \(X\), \(\varphi:B_X\to B_X\) a holomorphic self-map on \(B_X\) and \(C_\varphi\) the associated composition operator defined by \(C_\varphi(f) = f \circ \varphi\). Let \(H(B_X)\) be the space of all holomorphic functions \(f:B_X\to\mathbb C\). The aim of this paper is to study when the composition operator \(C_\varphi\) is power bounded, topologizable, and (uniform) mean ergodic on \(H(B_X)\) endowed with the compact-open topology, on the space \(H_b(B_X)\) of holomorphic functions of bounded type endowed with the topology of uniform convergence on \(B_X\)-bounded sets, as well as on \(H^\infty(B_X)\) the space of bounded functions on \(B_X\) endowed with the sup-norm. The obtained main result for \(H(B_X)\) is as follows.
The following statements are equivalent:
\begin{itemize}
\item[(i)] \(\varphi\) has stable orbits on \(B_X\).
\item[(ii)] \(C_\varphi:H(B_X)\to H(B_X)\) is power bounded.
\item[(iii)] \((\frac{1}{n}C_\varphi^n)_n\) is equiocontinuous in \(L(H(B_X))\).
\item[(iv)] \(C_\varphi:H(B_X)\to H(B_X)\) is topologizable.
\end{itemize}
A similar result holds for \(H_b(B_X)\). The authors also treat the above mentioned settings when \(X\) is a Hilbert space. Moreover, they provide several enlightening examples.
Reviewer: Mikael Lindström (Åbo)A class of operator-related composition operators from the Besov spaces into the Bloch spacehttps://zbmath.org/1521.470482023-11-13T18:48:18.785376Z"Zhu, Xiangling"https://zbmath.org/authors/?q=ai:zhu.xiangling"Abbasi, Ebrahim"https://zbmath.org/authors/?q=ai:abbasi.ebrahim"Ebrahimi, Ali"https://zbmath.org/authors/?q=ai:ebrahimi.aliSummary: The boundedness and the compactness of a class of operator-related composition operators from the Besov spaces into the Bloch space are given in this work.Complex symmetric Toeplitz operators on the unit polydiskhttps://zbmath.org/1521.470492023-11-13T18:48:18.785376Z"Dong, Xingtang"https://zbmath.org/authors/?q=ai:dong.xingtang"Gao, Yongxin"https://zbmath.org/authors/?q=ai:gao.yongxin"Hu, Qiuju"https://zbmath.org/authors/?q=ai:hu.qiujuSummary: In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators \(W_{\psi,\varphi}J\) are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators \(T_{z^{p\overline{z}q}}\) with \(p=Vq\) for some symmetric permutation matrix \(V\). Surprisingly, these operators \(T_{z^{Vq\overline{z}q}}\) are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk.Noncompactness of Toeplitz operators between abstract Hardy spaceshttps://zbmath.org/1521.470502023-11-13T18:48:18.785376Z"Karlovich, Alexei"https://zbmath.org/authors/?q=ai:karlovych.oleksiyExtending the classical result by Brown and Halmos on compactness of Toeplitz operators on the Hardy space over the unit circle \(\mathbb{T}\), \textit{K. Leśnik} [Stud. Math. 249, No. 2, 163--192 (2019; Zbl 1521.47052)] proved that a Toeplitz operator, acting between abstract Hardy spaces \(H[X]\) and \(H[Y]\) built upon possibly different rearrangement-invariant Banach function spaces \(X\) and \(Y\) over \(\mathbb{T}\) and such that \(Y\) has nontrivial Boyd indices, is compact if and only if its symbol is identically zero a.e. on \(\mathbb{T}\). The paper provides further extensions of this result for much more general spaces \(X\) and \(Y\). In particular, it is shown that there are no nontrivial compact Toeplitz operators on the Hardy space \(H^1 = H[L^1]\), although \(L^1\) has trivial Boyd indices. The author's approach is based on the use of the Kuratowski measure of noncompactness.
Reviewer: Nikolaj L. Vasilevskij (Ciudad de México)Generalized Crofoot transform and applicationshttps://zbmath.org/1521.470512023-11-13T18:48:18.785376Z"Khan, Rewayat"https://zbmath.org/authors/?q=ai:khan.rewayat"Farooq, Aamir"https://zbmath.org/authors/?q=ai:farooq.aamirSummary: Matrix-valued asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These are the generalization of matrix-valued truncated Toeplitz operators. In this article, we describe symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator. We also use generalized Crofoot transform to find a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators \(\mathcal{T}(\Theta_1, \Theta_2)\) and \(\mathcal{T}(\Theta_1^\prime, \Theta_2^\prime)\).Toeplitz and Hankel operators between distinct Hardy spaceshttps://zbmath.org/1521.470522023-11-13T18:48:18.785376Z"Leśnik, Karol"https://zbmath.org/authors/?q=ai:lesnik.karolSummary: The paper deals with Toeplitz and Hankel operators acting between distinct Hardy type spaces over the unit circle \(\mathbb{T}\). We characterize possible symbols of such operators and prove general versions of the Brown-Halmos theorem and the Nehari theorem. A~lower bound for the Kuratowski measure of noncompactness of a Toeplitz operator is also found. Our approach allows handling Hardy spaces associated with arbitrary rearrangement invariant spaces, but the main part of the results are new even for the classical \(H^p\) spaces.Moment maps of abelian groups and commuting Toeplitz operators acting on the unit ballhttps://zbmath.org/1521.470532023-11-13T18:48:18.785376Z"Quiroga-Barranco, Raúl"https://zbmath.org/authors/?q=ai:quiroga-barranco.raul"Sánchez-Nungaray, Armando"https://zbmath.org/authors/?q=ai:sanchez-nungaray.armandoSummary: We prove that to every connected abelian subgroup \(H\) of the biholomorphisms of the unit ball \(\mathbb{B}^n\) we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative \(C^\ast\)-algebra on every weighted Bergman space. These symbols are of the form \(a(z) = f (\mu^H(z))\), where \(\mu^H\) is the moment map for the action of \(H\) on \(\mathbb{B}^n\). We show that, for this construction, if \(H\) is a maximal abelian subgroup, then the symbols introduced are precisely the \(H\)-invariant symbols. We provide the explicit computation of moment maps to obtain special sets of symbols described in terms of coordinates. In particular, it is proved that our symbol sets have as particular cases all symbol sets from the current literature that yield Toeplitz operators generating commutative \(C^\ast\)-algebras on all weighted Bergman spaces on the unit ball \(\mathbb{B}^n\). Furthermore, we exhibit examples that show that some of the symbol sets introduced in this work have not been considered before. Finally, several explicit formulas for the corresponding spectra of the Toeplitz operators are presented. These include spectral integral expressions that simplify the known formulas for maximal abelian subgroups for the unit ball.Eigenvalue asymptotics for a class of multi-variable Hankel matriceshttps://zbmath.org/1521.470542023-11-13T18:48:18.785376Z"Tantalakis, Christos Panagiotis"https://zbmath.org/authors/?q=ai:tantalakis.christos-panagiotisSummary: A one-variable Hankel matrix \(H_a\) is an infinite matrix \(H_a = [a(i+j)]_{i, j\geq 0}\). Similarly, for any \(d \geq 2\), a \(d\)-variable Hankel matrix is defined as \(H_{\mathbf{a}} = [\mathbf{a}(\mathbf{i} + \mathbf{j})]\), where \(\mathbf{i} = (i_1, \dots, i_d)\) and \(\mathbf{j} = (j_1, \dots, j_d)\), with \(i_1, \dots, i_d, j_1, \dots, j_d \geq 0\). For \(\gamma > 0\), \textit{A. Pushnitski} and \textit{D. Yafaev} [Int. Math. Res. Not. 2015, No. 22, 11861--11886 (2015; Zbl 1338.47025)] proved that the eigenvalues of the compact one-variable Hankel matrices \(H_a\) with \(a(j) = j^{-1}(\log j)^{-\gamma}\), for \(j \geq 2\), obey the asymptotics \(\lambda_n(H_a) \sim C_\gamma n^{-\gamma}\), as \(n\to+\infty\), where the constant \(C_\gamma\) is calculated explicitly. This article presents the following \(d\)-variable analogue. Let \(\gamma > 0\) and \(a(j) = j^{-d}(\log j)^{-\gamma}\), for \(j \geq 2\). If \(\mathbf{a}(j_1, \dots, j_d) = a(j_1+\cdots+j_d)\), then \(H_{\mathbf{a}}\) is compact and its eigenvalues follow the asymptotics \(\lambda_n(H_{\mathbf{a}}) \sim C_{d, \gamma}n^{-\gamma}\), as \(n\to+\infty\), where the constant \(C_{d, \gamma}\) is calculated explicitly.Products of composition and differentiation between the fractional Cauchy spaces and the Bloch-type spaceshttps://zbmath.org/1521.470552023-11-13T18:48:18.785376Z"Hibschweiler, R. A."https://zbmath.org/authors/?q=ai:hibschweiler.rita-aFor \(\alpha > 0\), the space of \(F_{\alpha}\) of fractional Cauchy transforms is the family of functions of the form \(f(z) = \int_{\mathbb{T}}\frac{1}{(1-\overline{x}z)^{\alpha}}\,d\mu(x)\), \(|z| < 1\), where \(\mu \in \mathcal{M}\), the space of complex Borel measures on \( \mathbb{T} = \{ x \in \mathbb{C} : |x| =1 \}\) with norm \(\|f\|_{F_{\alpha}} = \inf \|\mu\|\), and \(\mu\) varies in \(\mathcal{M}\) for which the representation of \(f(z)\) holds. \(F_{\alpha}\) is a Banach space with the norm \(\|f\|_{F_{\alpha}}\). For \(\beta > 0\), the Bloch-type space \(B^{\beta}\) is the Banach space of functions analytic in the unit disk \(\mathbb{U}\) such that \(\sup_{z \in \mathbb{U}}(1-|z|^{\alpha})^{\beta}|f'(z)| < \infty\) with norm \(\|f\| _{B^{\beta}} = |f(0)|+\sup_{z \in \mathbb{U}} (1-|z|^{\alpha})^{\beta}|f'(z)|\). It is clear from the above definitions that \(F_{\alpha} \subseteq B^{\alpha +1}\), and there is a number \(C\) depending only on \(\alpha\) such that \(\|f\|_{B^{\beta}} \le C\|f\|_{F^{\alpha}}\) for all \(f \in F^{\alpha}\).
Let \(\phi\) be an analytic self-map of \(\mathbb{U}\). The products \(C_{\phi}D\) and \(DC_{\phi}\), where \(C_{\phi}\) and \(D\) are composition and differentiation operators, respectively, have been studied extensively by \textit{S.~Ohno} [Bull. Korean Math. Soc. 46, No. 6, 1135--1140 (2009; Zbl 1177.47044)]. \textit{S.-X. Li} and \textit{S. Stević} [J. Appl. Funct. Anal. 3, No. 3, 333--340 (2008; Zbl 1177.47043)] studied \(DC_{\phi}\) and \(C_{\phi}D\) between the weighted Bergman spaces and the Bloch-type spaces, and in [Rocky Mt. J. Math. 35, No. 3, 843--855 (2005; Zbl 1079.47031)], \textit{R. A. Hibschweiler} and \textit{N. Portnoy} studied these operators acting between Bergman spaces and Hardy spaces.
The paper under review investigates \(DC_{\phi}\) and \(C_{\phi}D\) acting between \(F_{\alpha}\) and \(B^{\beta}\) for \(\alpha , \beta > 0\). For \(\beta < 2\), the author gives a characterization for which \(DC_{\phi} : F_{\alpha} \rightarrow B^{\beta}\) and \(C_{\phi}D : F_{\alpha} \rightarrow B^{\beta}\) are bounded or compact. For \( \alpha >0\), \(0 < \beta < 2 \), the author proves that \(DC_{\phi} : F_{\alpha} \rightarrow B^{\beta}\) is compact iff \(DC_{\phi} : F_{\alpha} \rightarrow B^{\beta}\) is bounded iff \(\phi' \in B^{\beta}, \ \ \phi\phi' \in B^{\beta}\) and \(\|\phi\|_{\infty} < 1\).
For \( \alpha > 0\), \(0 < \beta < 1 \), it is shown that \(C_{\phi}D : F_{\alpha} \rightarrow B^{\beta}\) is compact iff \(C_{\phi}D : F_{\alpha} \rightarrow B^{\beta}\) is bounded iff \(\phi \in B^{\beta}\) and \(\|\phi\|_{\infty} < 1\).
Reviewer: Abebaw Tadesse (Langston)Unified approach to spectral properties of multipliershttps://zbmath.org/1521.470562023-11-13T18:48:18.785376Z"Lindström, Mikael"https://zbmath.org/authors/?q=ai:lindstrom.mikael"Miihkinen, Santeri"https://zbmath.org/authors/?q=ai:miihkinen.santeri"Norrbo, David"https://zbmath.org/authors/?q=ai:norrbo.davidThe main object of this paper is to extend a result by \textit{G.-F. Cao} et al. [J. Funct. Anal. 275, No. 5, 1259--1279 (2018; Zbl 1402.32007)] characterizing the spectrum and essential spectrum of the multiplication operator \(M_u\) on the Hardy-Sobolev Hilbert space, to Bergman-Sobolev and Bloch-type spaces of the open unit ball \({\mathbb B}^n\) of \({\mathbb C}^n\).
Let \((X({\mathbb B}^n), \|\cdot\|_X)\) and \((Y({\mathbb B}^n),\|\cdot\|_Y)\) be Banach spaces of analytic functions on the unit ball in \({\mathbb C}^n\), \({\mathbb B}^n\), containing the constants. Assume that the spaces \(X({\mathbb B}^n)\) and \(Y({\mathbb B}^n)\) satisfy the three conditions below:
\begin{itemize}
\item[1.] The topologies induced by \(\|\cdot\|_X\) and \(\|\cdot\|_Y\) are both finer than the compact-open topology \(\tau_0\). In particular, for every \(z\in {\mathbb B}^n\), the evaluation functional \(\delta_z(f)=f(z)\) is a bounded linear functional on both \(X({\mathbb B}^n)\) and \(Y({\mathbb B}^n)\).
\item[2.] For some \(N\in {\mathbb N}\),
\[
\|f\|_X\simeq |f(0)|+\|R^N f\|_X, \quad f\in X({\mathbb B}^n).
\]
\item[3.] \(H^\infty({\mathbb B}^n) \subset M(Y({\mathbb B}^n))\), the space of pointwise multipliers on \(Y({\mathbb B}^n)\), given by
\[
M(Y({\mathbb B}^n))=\{ u \in H({\mathbb B}^n);\, uf\in Y({\mathbb B}^n),\, f\in Y({\mathbb B}^n)\}.
\]
\end{itemize}
These spaces include the Bergman-Sobolev spaces and Bloch-type spaces on the unit ball.
The main result in this paper is the following.
Theorem. Assume that conditions (i) to (iii) are satisfied and let \(M_u : X({\mathbb B}^n) \to X({\mathbb B}^n)\) be the multiplication operator generated by \(u\in M(X({\mathbb B}^n))\). Then the spectrum of \(M_u\) is given by \(\sigma( M_u)=\overline{u({\mathbb B}^n)}\).
Reviewer: Carme Cascante (Barcelona)Persistence of point spectrum for perturbations of one-dimensional operators with discrete spectrahttps://zbmath.org/1521.470572023-11-13T18:48:18.785376Z"de Oliveira, César R."https://zbmath.org/authors/?q=ai:de-oliveira.cesar-r"Pigossi, Mariane"https://zbmath.org/authors/?q=ai:pigossi.marianeSummary: Sufficient conditions are given, for the preservation of the pure point spectrum, as well as dynamical localization properties, of autonomous and time-periodic perturbations of self-adjoint operators in \(\ell^2(\mathbb{Z})\) with simple pure point spectra whose eigenvalues have no accumulation point.
For the entire collection see [Zbl 1461.35006].The spectra of the generalized difference operators on the spaces of convergent serieshttps://zbmath.org/1521.470582023-11-13T18:48:18.785376Z"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiro"El-Shabrawy, Saad R."https://zbmath.org/authors/?q=ai:el-shabrawy.saad-rSummary: An investigation is made on the spectrum and fine spectrum of the generalized difference operator \(\Delta_{ab}\) on the space \(cs\) of convergent series. Some related results, concerning the spectrum of the adjoint operator \((\Delta_{ab})^*=\Delta^{ab}\) on the space \(bv\) of bounded variation sequences, are also derived. Further, some special cases of the main results are investigated deeper.Extensions of the arithmetic-geometric means and Young's norm inequalities to accretive operators, with applicationshttps://zbmath.org/1521.470592023-11-13T18:48:18.785376Z"Jocić, Danko R."https://zbmath.org/authors/?q=ai:jocic.danko-r"Krtinić, Đorđe"https://zbmath.org/authors/?q=ai:krtinic.dorde"Lazarević, Milan"https://zbmath.org/authors/?q=ai:lazarevic.milanSummary: If \(A,B \in \mathcal{B}(\mathcal{H})\) are normal accretive operators, \(X \in \mathcal{B}(\mathcal{H})\), \(0<\alpha <1\) and \(\Phi\) is a `symmetrically norming' (s.n.)\ function, we prove that
\[
\begin{aligned}
&|(A^{\ast}+A)^{1-\alpha}X(B+B^{\ast})^{\alpha}|^2\\
&\quad \leqslant \Gamma (2-2\alpha) \int_{[0,+\infty)} \mathrm{e}^{-tB^{\ast}}(B^{\ast}+B)^{\alpha} |AX+XB|^2 \\
&\qquad \times (B^{\ast}+B)^{\alpha} \mathrm{e}^{-tB} t^{2\alpha -1}\, \mathrm{d}t,
\end{aligned}
\]
\[
\begin{aligned}
&\|(A^{\ast}+A)^{1-\alpha}X(B+B^{\ast})^{\alpha}\|_{\Phi} \leqslant \Gamma (2-2\alpha) \Gamma (2\alpha) \|AX+XB\|_{\Phi}\quad\text{if } AX+XB \in \mathcal{C}_{\Phi}(\mathcal{H}).
\end{aligned}
\]
Let \(A,B,X \in \mathcal{B}(\mathcal{H})\), \(A \geqslant 0\), \(B \geqslant 0\), \(\eta, \theta \in \mathbb{R}\), \(\alpha \in (0,1)\) and \(\Phi\) be an s.n.\ function. If \(\mathrm{e}^{\mathrm{i}\eta} AX+\mathrm{e}^{\mathrm{i}\theta}XB \in \mathcal{C}_{\Phi} (\mathcal{H})\), then we have the following generalization of Young's norm inequality in [\textit{D. R. Jocić}, J. Funct. Anal. 218, No. 2, 318--346 (2005; Zbl 1083.46024)]:
\[
|\mathrm{e}^{\mathrm{i}\eta}+\mathrm{e}^{\mathrm{i}\theta}| \, \|A^{1-\alpha} XB^{\alpha}\}_{\Phi} \leqslant \sqrt{\Gamma (2-2\alpha)\Gamma (2\alpha)} \|\mathrm{e}^{\mathrm{i}\eta} AX + \mathrm{e}^{\mathrm{i}\theta} XB\|_{\Phi}.
\]
Various examples and applications of the obtained norm inequalities are also presented, including those related to the Heinz and Heron means and Zhan inequalities.On the automatic regularity of derivations from Riesz subalgebras of \(\mathcal{L}^r (X)\)https://zbmath.org/1521.470602023-11-13T18:48:18.785376Z"Blanco, A."https://zbmath.org/authors/?q=ai:blanco.anibal-m|blanco.andres|blanco.angela|blanco.armando|blanco.amalia|blanco.arielAuthor's abstract: We investigate the automatic regularity of bounded derivations from a Banach lattice algebra of regular operators $\mathcal{A}$ into a Banach $\mathcal{A}$-module with a Banach lattice structure compatible with the module operations.
Reviewer: Mohamed Ali Toumi (Bizerte)Around the closure of the set of commutators of idempotents in \(\mathcal{B}(\mathcal{H})\): biquasitriangularity and factorisationhttps://zbmath.org/1521.470612023-11-13T18:48:18.785376Z"Marcoux, Laurent W."https://zbmath.org/authors/?q=ai:marcoux.laurent-w"Radjavi, Heydar"https://zbmath.org/authors/?q=ai:radjavi.heydar"Zhang, Yuanhang"https://zbmath.org/authors/?q=ai:zhang.yuanhangSummary: In this paper, we continue our study of the norm-closure of the set \(\mathfrak{C}_{\mathfrak{E}}\) of bounded linear operators acting on a complex, infinite-dimensional, separable Hilbert space \(\mathcal{H}\) which may be expressed as the commutator of two idempotent operators. In particular, we identify which biquasitriangular operators belong to the norm-closure \(\mathrm{CLOS}(\mathfrak{C}_{\mathfrak{E}} )\) of \(\mathfrak{C}_{\mathfrak{E}} \), and we exhibit an index obstruction to membership in \(\mathrm{CLOS}(\mathfrak{C}_{\mathfrak{E}} )\). Finally, we consider factorisations of bounded linear operators on \(\mathcal{H}\) as sums and products of elements in \(\mathfrak{C}_{\mathfrak{E}}\) and related sets.On local automorphisms of some quantum mechanical structures of Hilbert space operatorshttps://zbmath.org/1521.470622023-11-13T18:48:18.785376Z"Gyenti, Bálint"https://zbmath.org/authors/?q=ai:gyenti.balint"Molnár, Lajos"https://zbmath.org/authors/?q=ai:molnar.lajosSummary: In this paper, we substantially strengthen several formerly obtained results stating that all 2-local automorphisms of certain quantum structures consisting of Hilbert space operators are necessarily automorphisms [\textit{L. Molnár}, Lett. Math. Phys. 58, No. 2, 91--100 (2001; Zbl 1002.46044); \textit{M. Barczy} and \textit{M. Tóth}, Rep. Math. Phys. 48, No. 3, 289--298 (2001; Zbl 1007.47014); \textit{L. Molnár}, J. Math. Anal. Appl. 479, No. 1, 569--580 (2019; Zbl 1490.46059)].Orthogonal sums in Kreĭn spaceshttps://zbmath.org/1521.470632023-11-13T18:48:18.785376Z"Rovnyak, James"https://zbmath.org/authors/?q=ai:rovnyak.jamesIn Krein spaces, the infinite sum of pairwise othogonal elements may cause more problems than we know from Hilbert spaces. Using elementary methods, the paper under review presents conditions such that the behaviour is similar to the Hilbert space case.
First of all, in a Krein space \({\mathcal H}\) with indefinite inner product \(\langle\cdot,\cdot\rangle\) and associated norm \(\| \cdot \|_{\mathcal H}\), a meaning is given to a sum \(\sum^s_{\lambda \in \Lambda} f_\lambda\) (in a natural way) where \(\Lambda\) is an arbitrary set of indices and \(\{f_\lambda\}_{\lambda \in \Lambda}\) is a family of vectors in \({\mathcal H}\) and, similarly, to \(\sum^s_{\lambda \in \Lambda} {\mathcal M}_\lambda\) where \(\{{\mathcal M}_\lambda\}_{\lambda \in \Lambda}\) is a family of subspaces of \({\mathcal H}\). If the \({\mathcal M}_\lambda\) are pairwise orthogonal and regular (i.e., closed and itself a Krein space with \(\langle\cdot,\cdot\rangle\)) with selfadjoint projections \(P_\lambda\), then the question arises whether \(\{P_\lambda f\}_{\lambda \in \Lambda}\) is summable for all \(f \in {\mathcal H}\).
It is shown that this is true if the operator norm \(\|\sum^s_{\lambda \in \Lambda_0} P_\lambda\|_{{\mathcal L}({\mathcal H})}\) has a fixed bound for all finite subsets \(\Lambda_0\) of \(\Lambda\) (or, equivalently, \(\|\sum^s_{\lambda \in \Lambda_0} P_\lambda f\|_{\mathcal H}\) has such a bound for each \(f \in {\mathcal H}\)). In this case, \(\sum^s_{\lambda \in \Lambda} {\mathcal M}_\lambda\) is a regular subspace and coincides with the closed span of \(\{{\mathcal M}_\lambda\}_{\lambda \in \Lambda}\) and the selfadjoint projection \(P\) onto this space allows the representation \(Pf = \sum^s_{\lambda \in \Lambda} P_\lambda f\) for all \(f \in {\mathcal H}\).
As a second result, it is shown that, again under a certain boundedness condition, the closed span is regular and isomorphic to a so-called external orthogonal sum \(\bigoplus_{\lambda \in \Lambda} {\mathcal M}_\lambda\) (using the fundamental decompositions of each component \({\mathcal M}_\lambda = {\mathcal M}^+_\lambda \oplus {\mathcal M}^-_\lambda\) and summing up the ``\(+\)'' and the ``\(-\)'' parts separately).
Finally, generalizations are obtained to the case of ``pseudo-regular'' subspaces (i.e., the orthogonal sum of a regular and a neutral subspace).
Reviewer: Andreas Fleige (Dortmund)On the lattice properties of almost L-weakly and almost M-weakly compact operatorshttps://zbmath.org/1521.470642023-11-13T18:48:18.785376Z"Akay, Barış"https://zbmath.org/authors/?q=ai:akay.baris"Gök, Ömer"https://zbmath.org/authors/?q=ai:gok.omer\textit{P. Meyer-Nieberg} [Math. Z. 138, 145--159 (1974; Zbl 0291.47020)] introduced L-weakly and M-weakly compact operators, which -- in contrast to the class of compact operators -- satisfy the domination property. In other words, if \(S,T:E\to F\) are operators between Banach lattices \(E\) and \(F\) such that \(0\le S\le T\) and \(T\) is L-weakly compact (respectively, M-weakly compact), then \(S\) is L-weakly compact (respectively, M-weakly compact).
As a generalization, \textit{K. Bouras} et al. [Positivity 22, No. 5, 1433--1443 (2018; Zbl 1496.46005)] introduced the classes of almost L-weakly and almost M-weakly compact operators. The paper under review shows that these two classes of operators also satisfy the domination property.
Furthermore, conditions are given for the linear span of the positive almost L-weakly compact (respectively, almost M-weakly compact) operators to be a Dedekind complete Banach lattice under the regular norm and for those Banach lattice to have an order continuous norm.
Reviewer: Sahiba Arora (Dresden)\(A\)-numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applicationshttps://zbmath.org/1521.470652023-11-13T18:48:18.785376Z"Bhunia, Pintu"https://zbmath.org/authors/?q=ai:bhunia.pintu"Feki, Kais"https://zbmath.org/authors/?q=ai:feki.kais"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallolLet \(A\) be a positive operator in \(\mathbb{B}(\mathscr{H})\), where \(\mathscr{H}\) is a complex Hilbert space. In this paper, the authors introduce the notion of \(A\)-numerical radius orthogonality for \(A\)-bounded operators. More precisely, for \(T , S\in \mathbb{B}_{A^{\frac{1}{2}}}(\mathscr{H}) \), they show that \(T\) is \(A\)-numerical radius orthogonal to \(S\) if and only if for each \(\beta \in [0, 2\pi)\), there exists a sequence of \(A\)-unit vectors \(\{x_k^{\beta}\}\) in \(\mathscr{H}\) such that
\[\lim_n | \langle Tx_n^{\beta}, x_n^{\beta}\rangle_A|=\omega_A(T) \text{ and } \Re \{e^{i\beta} \langle x_n^{\beta}, Tx_n^{\beta}\rangle_A \langle Sx_n^{\beta}, x_n^{\beta}\rangle_A\}\geq 0.\]
Moreover, a characterization of the \(A\)-numerical radius parallelism of \(A\)-rank one operators is established. Their results cover and extend the works in [\textit{M. Mehrazin} et al., Bull. Iran. Math. Soc. 46, No. 3, 821--829 (2020; Zbl 1521.47010)]. They also obtain some inequalities for the \(A\)-numerical radius of semi-Hilbertian space operators as an application of the \(A\)-numerical radius orthogonality and parallelism.
Reviewer: Maryam Amyari (Mashhad)Correction to: ``\(A\)-numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications''https://zbmath.org/1521.470662023-11-13T18:48:18.785376Z"Bhunia, Pintu"https://zbmath.org/authors/?q=ai:bhunia.pintu"Feki, Kais"https://zbmath.org/authors/?q=ai:feki.kais"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallolFrom the text: In the original article [the authors, ibid. 47, No. 2, 435--457 (2021; Zbl 1521.47065)], during the final typesetting stage the equation in the proof of the Theorem 3.5 was published incorrectly.Discrete bilinear operators and commutatorshttps://zbmath.org/1521.470672023-11-13T18:48:18.785376Z"Bényi, Árpád"https://zbmath.org/authors/?q=ai:benyi.arpad"Oh, Tadahiro"https://zbmath.org/authors/?q=ai:oh.tadahiroSummary: We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the Hörmander classes \(BS^{\omega}_{1, 0}\). In particular, we investigate their relation to discrete analogs of the bilinear Calderón-Zygmund singular integral operators and show compactness of their commutators.Essential norm of weighted differentiation composition operators on Bergman spaces with admissible weightshttps://zbmath.org/1521.470682023-11-13T18:48:18.785376Z"Farooq, Shayesta"https://zbmath.org/authors/?q=ai:farooq.shayesta"Sharma, Ajay K."https://zbmath.org/authors/?q=ai:sharma.ajay-kumar.1|sharma.ajay-k"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-aymanSummary: In this paper, we obtain some inequalities involving norm and essential norm of weighted differentiation composition on Bergman spaces with admissible Békollé weights.On Stević-Sharma operators from the minimal Möbius invariant space into Zygmund-type spaceshttps://zbmath.org/1521.470692023-11-13T18:48:18.785376Z"Guo, Zhitao"https://zbmath.org/authors/?q=ai:guo.zhitao"Liu, Linlin"https://zbmath.org/authors/?q=ai:liu.linlin"Zhao, Xianfeng"https://zbmath.org/authors/?q=ai:zhao.xianfengSummary: The boundedness, essential norm and compactness of Stević-Sharma operators from the minimal Möbius invariant space into Zygmund-type spaces are investigated in this paper.On a new product-type operator on the unit ballhttps://zbmath.org/1521.470702023-11-13T18:48:18.785376Z"Stević, Stevo"https://zbmath.org/authors/?q=ai:stevic.stevoSummary: Let \(m\in\mathbb{N}\), \(u_j\), \(j = \overline{1, m}\), be holomorphic functions on the open unit ball \(\mathbb{B}\subset\mathbb{C}^n\), \(\varphi\) be a holomorphic self-map of \(\mathbb{B}\), and \(D_l\) be the partial derivative operator in the \(l\)th variable \(l\in\{1, 2, \dots, n\}\). We introduce here the following polynomial differentiation composition operator
\[
P^m_{D, \varphi}f := \sum\limits_{j=1}^m u_j C_\varphi D_{l_j}\cdots D_{l_1}f
\]
and give some necessary and sufficient conditions for the boundedness and compactness of the operator from the logarithmic Bloch spaces to weighted-type spaces of holomorphic functions on \(\mathbb{B}\).Random Hamiltonians with arbitrary point interactions in one dimensionhttps://zbmath.org/1521.470712023-11-13T18:48:18.785376Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Helman, Mark"https://zbmath.org/authors/?q=ai:helman.mark"Kesten, Jacob"https://zbmath.org/authors/?q=ai:kesten.jacob"Sukhtaiev, Selim"https://zbmath.org/authors/?q=ai:sukhtaiev.selimSummary: We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schrödinger operators with Bernoulli-type random singular potential and singular density.Long-term analysis of positive operator semigroups via asymptotic dominationhttps://zbmath.org/1521.470742023-11-13T18:48:18.785376Z"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochen"Wolff, Manfred P. H."https://zbmath.org/authors/?q=ai:wolff.manfredSummary: We consider positive operator semigroups on ordered Banach spaces and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup \(\mathcal{T}\) are inherited by other semigroups which are asymptotically dominated by \(\mathcal{T}\). Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.Positive semigroups and perturbations of boundary conditionshttps://zbmath.org/1521.470752023-11-13T18:48:18.785376Z"Gwiżdż, Piotr"https://zbmath.org/authors/?q=ai:gwizdz.piotr"Tyran-Kamińska, Marta"https://zbmath.org/authors/?q=ai:tyran-kaminska.martaSummary: We present a generation theorem for positive semigroups on an \(L^1\) space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.Compensated compactness: continuity in optimal weak topologieshttps://zbmath.org/1521.470762023-11-13T18:48:18.785376Z"Guerra, André"https://zbmath.org/authors/?q=ai:guerra.andre"Raiţă, Bogdan"https://zbmath.org/authors/?q=ai:raita.bogdan"Schrecker, Matthew R. I."https://zbmath.org/authors/?q=ai:schrecker.matthew-r-iAuthors' abstract: For \(l\)-homogeneous linear differential operators \(\mathcal{A}\) of constant rank, we study the implication
\begin{align*}
\left.
\begin{matrix}
v_j \rightharpoonup v \text{ in } X \\
\mathcal{A} v_j \to \mathcal{A} v \text{ in } W^{- l} Y
\end{matrix}
\right\} \Rightarrow F(v_j) \rightsquigarrow F(v) \text{ in } Z,
\end{align*}
where \(F\) is an \(\mathcal{A}\)-quasiaffine function and \(\rightsquigarrow\) denotes an appropriate type of weak convergence. Here, \(Z\) is a local \(L^1\)-type space, either the space \(\mathscr{M}\) of measures, or \(L^1\), or the Hardy space \(\mathscr{H}^1\); \(X, Y\) are \(L^p\)-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of \(X, Y, Z\) are sharp. Analogous statements are also given in the case when \(F(v)\) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove \(\mathscr{H}^p\)-bounds for the sequence \((F (v_j))_j\), for appropriate \(p < 1\), and new convergence results in the dual of Hölder spaces when \((v_j)\) is \(\mathcal{A}\)-free and lies in a suitable negative order Sobolev space \(W^{- \beta, s}\). The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
Reviewer: Mohammed El Aïdi (Bogotá)On some integral operators appearing in scattering theory, and their resolutionshttps://zbmath.org/1521.470812023-11-13T18:48:18.785376Z"Richard, Serge"https://zbmath.org/authors/?q=ai:richard.serge"Umeda, Tomio"https://zbmath.org/authors/?q=ai:umeda.tomioSummary: We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered.
For the entire collection see [Zbl 1461.35006].Lie centralizers at zero products on a class of operator algebrashttps://zbmath.org/1521.471152023-11-13T18:48:18.785376Z"Ghahramani, Hoger"https://zbmath.org/authors/?q=ai:ghahramani.hoger"Jing, Wu"https://zbmath.org/authors/?q=ai:jing.wuSummary: Let \(\mathcal{A}\) be an algebra. In this paper, we consider the problem of determining a linear map \(\psi\) on \(\mathcal{A}\) satisfying \(a,b\in\mathcal{A}\), \(ab=0\Longrightarrow\psi([a,b])=[\psi(a),b]\, (C1)\) or \(ab=0\Longrightarrow\psi([a,b])=[a,\psi(b)]\,(C2)\). We first compare linear maps satisfying \((C1)\) or \((C2)\), commuting linear maps, and Lie centralizers with a variety of examples. In fact, we see that linear maps satisfying \((C1), (C2)\) and commuting linear maps are different classes of each other. Then, we introduce a class of operator algebras on Banach spaces such that if \(\mathcal{A}\) is in this class, then any linear map on \(\mathcal{A}\) satisfying \((C1)\) (or \((C2))\) is a commuting linear map. As an application of these results, we characterize Lie centralizers and linear maps satisfying \((C1)\) (or \((C2))\) on nest algebras.A Dixmier trace formula for the density of stateshttps://zbmath.org/1521.471172023-11-13T18:48:18.785376Z"Azamov, N."https://zbmath.org/authors/?q=ai:azamov.nurulla-a"McDonald, E."https://zbmath.org/authors/?q=ai:mcdonald.edward-a"Sukochev, F."https://zbmath.org/authors/?q=ai:sukochev.fedor-a"Zanin, D."https://zbmath.org/authors/?q=ai:zanin.dmitriy-vSummary: A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics, there is another celebrated measure associated with such operators -- the density of states. In this paper, we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.Diagrams and harmonic maps, revisitedhttps://zbmath.org/1521.580072023-11-13T18:48:18.785376Z"Pacheco, Rui"https://zbmath.org/authors/?q=ai:pacheco.rui"Wood, John C."https://zbmath.org/authors/?q=ai:wood.john-cIn the paper under review, the authors study and apply the finiteness criterion they have developed in collaboration with A. Aleman to extend many known results initially established for harmonic maps from the \(2\)-sphere into a Grassmannian
to harmonic maps from an arbitrary Riemann surface into a finite uniton number.
These results include: the description of harmonic maps of finite uniton number from any Riemann surface into a complex Grassmannian \(G_{2}(\mathbb{C}^{n})\). Noting that this extension relies on a new theory of nilpotent cycles. Also, a constancy result is developed showing that a harmonic map from a torus to a complex Grassmannian which is simultaneously of
finite uniton number and finite type is constant. Besides, authors generalize how harmonic maps of finite uniton number
can be constructed explicitly from holomorphic data from any Riemann surface instead of the \(2\)-sphere, using new methods.
Reviewer: Hiba Bibi (Tours)Time evolution for quantum systems with a dynamical Hilbert spacehttps://zbmath.org/1521.810082023-11-13T18:48:18.785376Z"Chou, Hsiang Shun"https://zbmath.org/authors/?q=ai:chou.hsiang-shun(no abstract)Newton's identities and positivity of trace class integral operatorshttps://zbmath.org/1521.810162023-11-13T18:48:18.785376Z"Homa, G."https://zbmath.org/authors/?q=ai:homa.gabor"Balka, R."https://zbmath.org/authors/?q=ai:balka.richard"Bernád, J. Z."https://zbmath.org/authors/?q=ai:bernad.j-z"Károly, M."https://zbmath.org/authors/?q=ai:karoly.m"Csordás, A."https://zbmath.org/authors/?q=ai:csordas.andrasSummary: We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.Quantum and classical dynamical semigroups of superchannels and semicausal channelshttps://zbmath.org/1521.810452023-11-13T18:48:18.785376Z"Hasenöhrl, Markus"https://zbmath.org/authors/?q=ai:hasenohrl.markus"Caro, Matthias C."https://zbmath.org/authors/?q=ai:caro.matthias-cSummary: Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the Gorini-Kossakowski-Lindblad-Sudarshan form in the case of quantum channels. To derive the normal form, we exploit the relation between superchannels and semicausal completely positive maps, reducing the problem to finding a normal form for the generators of semigroups of semicausal completely positive maps. We derive a normal for these generators using a novel technique, which applies also to infinite-dimensional systems. Our work paves the way for a thorough investigation of semigroups of superchannels: Numerical studies become feasible because admissible generators can now be explicitly generated and checked. Analytic properties of the corresponding evolution equations are now accessible via our normal form.
{\copyright 2022 American Institute of Physics}Logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrödinger operators with smooth potentialshttps://zbmath.org/1521.810682023-11-13T18:48:18.785376Z"Jitomirskaya, Svetlana"https://zbmath.org/authors/?q=ai:jitomirskaya.svetlana-ya"Powell, Matthew"https://zbmath.org/authors/?q=ai:powell.matthew-jSummary: We present a method for obtaining power-logarithmic bounds on the growth of the moments of the position operator for one-dimensional ergodic Schrödinger operators. We use Bourgain's semialgebraic method to obtain such bounds for operators with multifrequency shift or skew-shift underlying dynamics with arithmetic conditions on the parameters.
For the entire collection see [Zbl 1506.46001].Dirac-Coulomb operators with infinite mass boundary conditions in sectorshttps://zbmath.org/1521.810742023-11-13T18:48:18.785376Z"Cassano, Biagio"https://zbmath.org/authors/?q=ai:cassano.biagio"Gallone, Matteo"https://zbmath.org/authors/?q=ai:gallone.matteo"Pizzichillo, Fabio"https://zbmath.org/authors/?q=ai:pizzichillo.fabioSummary: We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in the presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results, giving a description of the spectrum of these operators.
{\copyright 2022 American Institute of Physics}On the spectrum of the one-particle Schrödinger operator with point interactionhttps://zbmath.org/1521.810762023-11-13T18:48:18.785376Z"Kulzhanov, Utkir"https://zbmath.org/authors/?q=ai:kulzhanov.utkir"Muminov, Z. I."https://zbmath.org/authors/?q=ai:muminov.zahriddin-i"Ismoilov, Golibjon"https://zbmath.org/authors/?q=ai:ismoilov.golibjonSummary: We consider a one-dimensional one-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points \(\pm x_0\). The corresponding Schrödinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of the eigenvalue of the Schrödinger operator is studied. The main results of the work are based on the study of the operator extension spectrum of the operator \(h\).Localization for random quasi-one-dimensional modelshttps://zbmath.org/1521.820142023-11-13T18:48:18.785376Z"Boumaza, H."https://zbmath.org/authors/?q=ai:boumaza.hakimSummary: In this Review Article, we review the results of Anderson localization for different random families of operators that enter the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical points of view. From the Anderson-Bernoulli conjecture in dimension 2, we justify the introduction of quasi-one-dimensional models. Then, we present different types of these models: the Schrödinger type in the discrete and continuous cases, the unitary type, the Dirac type, and the point interaction type. We present tools coming from the study of dynamical systems in dimension one: the transfer matrix formalism, the Lyapunov exponents, and the Furstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schrödinger type involving only geometric and algebraic properties of the Furstenberg group. Then, we review results of localization, first for Schrödinger-type models and then for unitary type models. Each time, we reduce the question of localization to the study of the Furstenberg group and show how to use more and more refined algebraic criteria to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schrödinger type include the case of Bernoulli randomness.
{\copyright 2023 American Institute of Physics}