<?xml version="1.0" encoding="utf-8"?> <feed xmlns="http://www.w3.org/2005/Atom"> <title type="text">Recent zbMATH articles in MSC 47B</title> <id>https://zbmath.org/atom/cc/47B</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/" /> <link href="https://zbmath.org/atom/cc/47B" rel="self" /> <generator>Werkzeug</generator> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Matrix analysis and entrywise positivity preservers</title> <id>https://zbmath.org/1552.15001</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.15001" /> <author> <name>&quot;Khare, Apoorva&quot;</name> <uri>https://zbmath.org/authors/?q=ai:khare.apoorva</uri> </author> <content type="text">This book presents a relevant account on matrix positivity and preservers. The text, that is divided in three parts, is based on a semester course taught in the years 2018 and 2019 by the author at the India Institute of Science. In Part I there is a matrix analysis introduction on positive semidefinite and Hankel T-N matrices. This is done in the first four chapters, where we have a comprehensive exposition of matrix analysis related to semi-positiveness. The text is self-contained and accessible to students and researchers not familiar with the subject. Chapters 5 to 9 contain more specific topics related to some classical results on entrywise positivity preservers, obtained mainly by Loewner, Guillot and fellows. In Chapter 10 are the related exercises with theoretical and practical questions. Part II contains the main results on positivity preservers, since the first works on this topic, developed by Menger, Fr茅chet, Bochner and Shoenberg in the first half of last century and later in more specific research, carried out by Loewner and Karlin and the next generation of researchers, which includes Fitzgerald, Horn, Micchelli and Pinkus among others. More recently the works of the author and collaborators, as Belton, Guillot and Putinar added new theory and techniques to the matter. The topics in Chapters 11 to 17 are: history and metric geometry, Loewner's determinant calculation, Schoenberg and Vasudeva theorems; Chapters 18 to 22 include: preservers of Loewner positivity, monotonicity and convexity on kernels, functions outside of forbidden diagonal blocks, Boas-Widder theorem on functions with positive differences, Menger theorem and Euclidean distance geometry; in Chapter 23 some exercises are collected. Part III explores entrywise functions which preserve positivity in a fixed dimension and according to the author, this is an interesting topic for research, which is still an open problem in general. This part mainly relies on two works, one by \textit{A. Belton} et al. [Adv. Math. 298, 325--368 (2016; Zbl 1339.15016)] and the second by \textit{A. Khare} and \textit{T. Tao} [Am. J. Math. 143, No. 6, 1863--1929 (2021; Zbl 1487.30003)]. The topics in Chapters 24 and 25 are: entrywise and rank-1 matrices polynomial preservers; Chapters 26 and 27 include: first-order approximation and leading term of Schur polynomials, exact quantitative bound and monotonicity of Schur polynomials; Chapter 28 presents polynomial preservers on matrices; Chapter 29 is about definitions of Cauchy-Littlewood of Schur polynomials. Exercises are in Chapter 30. The book contains extensive reference and bibliographic notes for each part. Reviewer: Edgar Pereira (Natal)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">On the uniqueness and computation of commuting extensions</title> <id>https://zbmath.org/1552.15018</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.15018" /> <author> <name>&quot;Koiran, Pascal&quot;</name> <uri>https://zbmath.org/authors/?q=ai:koiran.pascal</uri> </author> <content type="text">Summary: A tuple \((Z_1, \ldots, Z_p)\) of matrices of size \(r \times r\) is said to be a \textit{commuting extension} of a tuple \((A_1, \ldots, A_p)\) of matrices of size \(n \times n\) if the \(Z_i\) pairwise commute and each \(A_i\) sits in the upper left corner of a block decomposition of \(Z_i\) (here, \(r\) and \(n\) are two arbitrary integers with \(n &lt; r)\). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in \textit{V. Strassen}'s work [Linear Algebra Appl. 52--53, 645--685 (1983; Zbl 0514.15018)] on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called ``quantum Zeno dynamics.'' Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: \begin{itemize} \item[(i)] Theorems on the uniqueness of commuting extensions for three matrices or more. \item[(ii)] Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to \(r = 4n/3\), and are apparently the first provably efficient algorithms for this problem applicable beyond \(r = n+1\). \item[(iii)] A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices. \end{itemize}</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Functional identities on Banach algebras and matrix rings</title> <id>https://zbmath.org/1552.16016</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.16016" /> <author> <name>&quot;Luo, Kaijia&quot;</name> <uri>https://zbmath.org/authors/?q=ai:luo.kaijia</uri> </author> <author> <name>&quot;Li, Jiankui&quot;</name> <uri>https://zbmath.org/authors/?q=ai:li.jiankui</uri> </author> <content type="text">Let \(A\) be a unital Banach algebra and \(M\) a unital \(A\)-bimodule. Assume \(\delta\) and \(\tau\colon A\to M\) are additive maps. The main theorem (theorem 1) of the paper under review states that \(a^{-1}\delta(a) + \delta(a) a^{-1} + a\tau(a^{-1}) + \tau(a^{-1})a =0\) for every invertible \(a\in A\) if and only if \(\delta(x) = d(x)+x\delta(1)\) and \(\tau(x)= d(x)-\delta(1)x\), for every \(x\in A\). Here \(d\) is a Jordan derivation of \(A\). The authors also study additive maps \(\delta\), \(\tau\colon A\to M\) with the property \(\delta(a)b+ a\tau(b) = m\) for every \(a\), \(b\in A\) such that \(ab=n\in A\) is invertible, and \(m\in M\) is fixed. Reviewer: Plamen Koshlukov (Campinas)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents</title> <id>https://zbmath.org/1552.16039</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.16039" /> <author> <name>&quot;Kawa, Ab Hamid&quot;</name> <uri>https://zbmath.org/authors/?q=ai:kawa.ab-hamid</uri> </author> <author> <name>&quot;Hasan, S. N.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:hasan.s-n</uri> </author> <author> <name>&quot;Wani, Bilal Ahmad&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wani.bilal-ahmad</uri> </author> <content type="text">Summary: Suppose \(\mathcal{R}\) is a non-zero unital associative ring with a nontrivial idempotent ``\(e\)''. In this paper, we prove that under some mild conditions every multiplicative jordan \(n\)-higher derivations on \(\mathcal{R}\) is additive. Moreover, at the end of the paper, we have presented some applications of multiplicative Jordan \(n\)-higher derivations on triangular rings, nest algebra, upper triangular block matrix algebra, prime rings, von Neumann algebras.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Characterizations of Lie centralizers of triangular algebras</title> <id>https://zbmath.org/1552.16040</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.16040" /> <author> <name>&quot;Liu, Lei&quot;</name> <uri>https://zbmath.org/authors/?q=ai:liu.lei.26</uri> </author> <author> <name>&quot;Gao, Kaitian&quot;</name> <uri>https://zbmath.org/authors/?q=ai:gao.kaitian</uri> </author> <content type="text">Summary: Let \(\mathcal{A}\) be an unital algebra over the complex field \(\mathbb{C}\). A linear map \(\phi\) from \(\mathcal{A}\) into itself is called a Lie centralizer at a given point \(G\in \mathcal{A}\) if \(\phi ([S,T])=[S,\phi (T)]\) for all \(S, T\in \mathcal{A}\) with \(ST = G\). The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Limit formulas for the trace of the functional calculus of quantum channels for \(\mathrm{SU}(2)\)</title> <id>https://zbmath.org/1552.22056</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.22056" /> <author> <name>&quot;van Haastrecht, Robin&quot;</name> <uri>https://zbmath.org/authors/?q=ai:van-haastrecht.robin</uri> </author> <content type="text">This paper extends foundational results by \textit{E. H. Lieb} and \textit{J. P. Solovej} [Acta Math. 212, No. 2, 379--398 (2014; Zbl 1298.81116)], providing new limit formulas for traces in the functional calculus of quantum channels associated with \(\mathrm{SU}(2)\) representations. The author broadens the scope of quantum channels to include all components of \(\mathrm{SU}(2)\) tensor decompositions, not only the leading component and establishes the trace limiting formula for \(\phi \in C([0,1])\): \[ \lim_{\nu \to \infty} \frac{1}{\nu} \operatorname{Tr}(\phi(\mathcal{T}_{\mu,k}^\nu(R_\mu^*(f)))) = \int_{\mathbb C} \phi(E_{\mu,k}(f)) \frac{dz}{\pi(1 + |z|^2)^2}, \] for any \(f \in C(\mathbb{CP}^1)\) with \(\int_{\mathbb{CP}^1} f(z) \frac{dz}{\pi(1 + |z|^2)^2} = 1\) and Toeplitz operator \(R_\mu^*(f) \geq 0\), where \( E_{\mu, k}(f) \) incorporates the Berezin transforms. Some additional related results are also presented. Reviewer: Tin Yau Tam (Reno)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions</title> <id>https://zbmath.org/1552.34106</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.34106" /> <author> <name>&quot;Sakhnovich, Alexander L.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:sakhnovich.alexander-l</uri> </author> <content type="text">Summary: We solve the inverse problems to recover Dirac systems on an interval or semiaxis from their spectral functions (matrix valued functions) for the case of locally square-integrable potentials. Direct problems in terms of spectral functions are treated as well. Moreover, we present necessary and sufficient conditions on the given distribution matrix valued function to be a spectral function of some Dirac system with a locally square-integrable potential. Interesting connections with Paley-Wiener sampling measures appear in the case of scalar spectral functions.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Quasidifferentiability of families of eigenelements of Perron-Frobenius operators</title> <id>https://zbmath.org/1552.37029</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.37029" /> <author> <name>&quot;Sarazhinski沫, D. S.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:sarazhinsky.d-s</uri> </author> <content type="text">(no abstract)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">On localization and homogeneity for analytic quasi-periodic Schr枚dinger operators with Gevrey perturbation</title> <id>https://zbmath.org/1552.37030</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.37030" /> <author> <name>&quot;Tao, Kai&quot;</name> <uri>https://zbmath.org/authors/?q=ai:tao.kai</uri> </author> <content type="text">Summary: It was shown in [\textit{J. Bourgain} and \textit{M. Goldstein}, Ann. Math. (2) 152, No. 3, 835--879 (2000; Zbl 1053.39035); \textit{D. Damanik} et al., J. Eur. Math. Soc. (JEMS) 20, No. 12, 3073--3111 (2018; Zbl 1478.47023)] that the non-perturbative Anderson localization and homogeneous spectrum hold for the quasi-periodic analytic Schr枚dinger operator in the positive Lyapunov exponent regime. In this paper, we prove that they are both stable under the Gevrey perturbation on the potential with the Diophantine frequency.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Orthogonal polynomials and operator theory. Open problems and applications</title> <id>https://zbmath.org/1552.37056</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.37056" /> <author> <name>&quot;Huertas, Edmundo J.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:huertas.edmundo-j</uri> </author> <author> <name>&quot;Marcell谩n, Francisco&quot;</name> <uri>https://zbmath.org/authors/?q=ai:marcellan-espanol.francisco</uri> </author> <content type="text">Summary: In this contribution we review first the connections between orthogonal polynomials on the real line, Jacobi matrices, their LU and UL factorizations, spectral measures, spectral transformations and Stieltjes functions as well as their application in the study of some integrable systems of the Toda hierarchy. In a second step, we analyze symmetric operators associated with a Sobolev type inner product. The connection between the Cholesky factorization of five diagonal matrices associated with such operators and the UL factorization of the square of a shifted Jacobi matrix associated with a Christoffel perturbation of the measure is pointed out. Finally, we deal with orthogonal polynomials associated with probability measures supported on the unit circle, Hessenberg and CMV matrices, spectral measures and their spectral transformations, Carath茅odory functions as well as their application in the analysis of Schur flows both from an algebraic and analytic way. For the entire collection see [Zbl 1548.32001].</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">On the eigenstructure of the \(q\)-Durrmeyer operators</title> <id>https://zbmath.org/1552.41010</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.41010" /> <author> <name>&quot;Yilmaz, 脰vg眉 G眉rel&quot;</name> <uri>https://zbmath.org/authors/?q=ai:yilmaz.ovgu-gurel</uri> </author> <content type="text">Summary: The purpose of this paper is to establish the eigenvalues and the eigenfunctions of both the \(q\)-Durrmeyer operators \(D_{n, q}\) and the limit \(q\)-Durrmeyer operators \(D_{\infty, q}\) introduced by \textit{V. Gupta} [Appl. Math. Comput. 197, No. 1, 172--178 (2008; Zbl 1142.41008)] in the case \(0 &lt; q &lt; 1\). All moments for \(D_{n, q}\) and \(D_{\infty, q}\) are provided. The coefficients for the eigenfunctions of the operators are explicitly derived and the eigenfunctions of these operators are illustrated by graphical examples.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Time-frequency analysis associated with multidimensional Hankel-Gabor transform</title> <id>https://zbmath.org/1552.42009</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.42009" /> <author> <name>&quot;Chana, Ahmed&quot;</name> <uri>https://zbmath.org/authors/?q=ai:chana.ahmed</uri> </author> <author> <name>&quot;Akhlidj, Abdellatif&quot;</name> <uri>https://zbmath.org/authors/?q=ai:akhlidj.abdellatif</uri> </author> <author> <name>&quot;Nafie, Noureddine&quot;</name> <uri>https://zbmath.org/authors/?q=ai:nafie.noureddine</uri> </author> <content type="text">Summary: The main crux of this paper is to introduce a new integral transform called the multidimensional Hankel-Gabor transform which generalizes the classical Gabor Fourier transform and to give some new results related to this transform as Plancherel's, Parseval's, inversion and Calder贸n's reproducing formulas. Next, we analyse the concentration of this transform on sets of finite measure and we give uncertainty principle for orthonormal sequences. Last, using the best approximations and the theory of reproducing kernels, we study the extremal functions related to this transform and we give an integral representation, best estimates of these functions on weighted Sobolev spaces.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">\(\Theta\)-type fractional Marcinkiewicz integral operators and their commutators on some spaces over RD-spaces</title> <id>https://zbmath.org/1552.42016</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.42016" /> <author> <name>&quot;Lu, Guanghui&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lu.guanghui</uri> </author> <author> <name>&quot;Tao, Wenwen&quot;</name> <uri>https://zbmath.org/authors/?q=ai:tao.wenwen</uri> </author> <content type="text">Summary: The aim of this paper is to establish the boundedness of an \(\theta\)-type fractional Marcinkiewicz integral \(\mathcal{M}_{q,\rho,\alpha,\theta}\) and its commutator \(\mathcal{M}_{q,\rho,\alpha,\theta,b}\) on weighted Lebesgue spaces \(L^p (\omega)\), weighted Morrey spaces \(\mathcal{M}^{p,\kappa}(\mu)\) and generalized weighted Morrey spaces \(\mathcal{L}^{p,\Phi}(\omega)\) over RD-spaces satisfying the doubling and reverse doubling conditions. Under assumption that the functions \(\omega\) and \(\Phi\) satisfy some certain conditions, the authors prove that the \(\mathcal{M}_{q,\rho,\alpha,\theta}\) is bounded from spaces \(L^p (\omega)\) into spaces \(L^p (\omega)\), bounded from spaces \(\mathcal{M}^{p,\kappa}(\omega)\) into spaces \(\mathcal{M}^{p,\kappa}(\omega)\), and it is also bounded from spaces \(\mathcal{L}^{p,\Phi}(\omega)\) into spaces \(\mathcal{L}^{p,\Phi}(\omega)\), where \(p\in (1,\infty)\), \(\kappa \in (0,1)\), \(\omega \in A_p (\mu)\) and \(\Phi (\cdot)\) is a non-decreasing and non-negative function defined on \((0,\infty)\). Furthermore, by establishing the sharp maximal estimate for the commutator \(\mathcal{M}_{q,\rho,\alpha,\theta,b}\) which is formed by \(b\in \mathrm{BMO}(\mu)\) and the \(\mathcal{M}_{q,\rho,\alpha,\theta}\), the boundedness of the \(\mathcal{M}_{q,\rho,\alpha,\theta,b}\) on spaces \(L^p (\omega)\), spaces \(\mathcal{M}^{p,\kappa}(\omega)\) and spaces \(\mathcal{L}^{p,\Phi}(\omega)\) is obtained, respectively.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">\(\ell^1\)-bounded sets</title> <id>https://zbmath.org/1552.42026</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.42026" /> <author> <name>&quot;Heil, Christopher&quot;</name> <uri>https://zbmath.org/authors/?q=ai:heil.christopher-e</uri> </author> <author> <name>&quot;Yu, Pu-Ting&quot;</name> <uri>https://zbmath.org/authors/?q=ai:yu.pu-ting</uri> </author> <content type="text">This paper introduces and studies the concepts of \(\ell^1\) bounded sets and \(\ell^1\) frame bounded sets in a separable Hilbert space \(H\). These studies may lead to new, interesting aspects of frame theory. A subset \(M\) of \(H\) is \(\ell^1\) frame bounded if there is a frame \(\{e_n\}\) in \(H\) such that \(\sup_{x\in M}\sum_n |\langle x,e_n\rangle|&lt;\infty.\) If the frame is actually a Riesz basis, \(M\) is simply called an \(\ell^1\) bounded set. While \(\ell^1\) bounded sets are always bounded, the unit sphere in an infinite dimensional space is not \(\ell^1\) bounded. It is not known whether the union (or, equivalently, the sum) of finitely many \(\ell^1\) bounded sets is again \(\ell^1\) bounded. The authors conjecture that this is not the case. They base the conjecture on the fact that they are able, on the contrary case, to draw several unexpected conclusions (Theorem 4.12), e.g. every Bessel sequence is \(\ell^1\) bounded and so is every \(\ell^1\) frame bounded set. It is also shown that \(\ell^1\) boundedness in a larger Hilbert space \(K\) implies \(\ell^1\) boundedness in \(H\) if it has finite codimension in \(K\). The first part of the paper discusses, for frames \(\mathcal E\), the subspaces \(H_{\mathcal E}:=\{x: \sup_{x\in \mathcal E}\sum_n |\langle x,e_n\rangle|&lt;\infty\}.\) It is proved that \(H_{\mathcal E}\) is dense when \(\mathcal E\) is a Riesz basis, while it can be \((0)\) for a frame \(\mathcal E\). The class of \(H_{\mathcal E}\), \(\mathcal E\) is a Riesz basis, and does not have a smallest element nor a largest element. Related to the conjecture mentioned earlier, the following conjecture is also proposed: The collection of \(H_{\mathcal E}\) is not closed under finite unions or finite sums. The last section considers \(p\)-convergent frames and it is shown that there is no absolutely convergent frame while \(p\)-convergent frames exist for \(1&lt;p&lt;2\). Reviewer: K. Parthasarathy (Chennai)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Gabor like structured frames in separable Hilbert spaces</title> <id>https://zbmath.org/1552.42029</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.42029" /> <author> <name>&quot;Thomas, Jineesh&quot;</name> <uri>https://zbmath.org/authors/?q=ai:thomas.jineesh</uri> </author> <author> <name>&quot;Namboothiri, N. M. M.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:namboothiri.n-m-m</uri> </author> <author> <name>&quot;Nambudiri, T. C. E.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:easwaran-nambudiri.t-c</uri> </author> <content type="text">This article explores structured frames in separable Hilbert spaces, emphasizing Pseudo \( B \)-Gabor frames. The authors delve into the distinction between traditional Gabor frames and the Pseudo \( B \)-Gabor frames, proposing a framework where invertible operators map from \( L^2(\mathbb{R}) \) into a Hilbert space \( \mathcal{H} \). The most interesting contribution appears in Theorem 3.7, which stipulates conditions under which a bounded linear operator \( S \) on \( \mathcal{H} \) qualifies as a Pseudo \( B \)-Gabor-like frame operator. The work defines \( S \) through the relation \( S = T T^\ast \), where \( T = B A \) and \( A \) commute with specific translation and modulation operators. The significance of the article lies in extending the framework of frame operators beyond classical Gabor systems. The discussion on the conditions for the existence of Parseval Pseudo \( B \)-Gabor frames and their properties, such as positivity and commutativity, adds depth to the literature. This study offers insights into generating new types of frames in \( \mathcal{H} \), maintaining a structure analogous to Gabor frames in \( L^2(\mathbb{R}) \). Reviewer: Pierluigi Vellucci (Roma)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Some properties of the weak topology in the space \(\mathfrak{L}_\infty\)</title> <id>https://zbmath.org/1552.46013</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.46013" /> <author> <name>&quot;Alekhno, E. A.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:alekhno.egor-a</uri> </author> <content type="text">(no abstract)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Weyl type theorems in Banach algebras and hyponormal elements in \(C^*\) algebras</title> <id>https://zbmath.org/1552.46025</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.46025" /> <author> <name>&quot;Wu, Zhenying&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wu.zhenying</uri> </author> <author> <name>&quot;Zeng, Qingping&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zeng.qingping</uri> </author> <author> <name>&quot;Zhang, Yunnan&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zhang.yunnan</uri> </author> <content type="text">Summary: It is established that the relationships between Weyl's theorem, Browder's theorem, generalized Weyl's theorem and generalized Browder's theorem in a semiprime Banach algebra \({\mathcal{A}}\). We prove that if the commutant of \(a \in{\mathcal{A}}\) contains a left or right injective quasinilpotent element, then \(f(a)\) satisfies Weyl's theorem for \(f\) belongs to \({\mathcal{H}}ol(a)\), the set of analytic functions on a neighborhood of \(\sigma (a)\). It is shown that the accumulation points of the spectrum of an element in \({\mathcal{A}}\) are invariant under any commuting perturbation \(f\) such that \(f^n \in \mathrm{soc}({\mathcal{A}})\) for some \(n \in{\mathbb{N}}\). This result provides a positive answer to Question~2.8 in [\textit{Q.-P. Zeng} et al., Linear Multilinear Algebra 64, No.~2, 247--257 (2016; Zbl 1336.47004)], and it is then applied to investigate the perturbations of Weyl's theorem and generalized Weyl's theorem. It is also shown that if \(a\) is a hyponormal element (that is, \(a^*a \geq aa^*\)) in a \(C^*\) algebra \({\mathcal{A}}\) and \(f \in{\mathcal{H}}ol(a)\), then \(f(a)\) satisfies Weyl's theorem. If additionally \({\mathcal{A}}\) is primitive then \(f(a)\) obeys generalized Weyl's theorem. We also consider some other interesting properties of hyponormal elements in \(C^*\) algebras, including simply polaroidness, topological divisor of zero, self-adjointness of the spectral projection with respect to \(\lambda \in \mathrm{iso}\sigma (a)\), and the spectral mapping theorem of the Weyl spectrum.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">New tensor products of \(\mathrm{C}^\ast\)-algebras and characterization of type I \(\mathrm{C}^\ast\)-algebras as rigidly symmetric \(\mathrm{C}^\ast\)-algebras</title> <id>https://zbmath.org/1552.46032</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.46032" /> <author> <name>&quot;Lee, Hun Hee&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lee.hun-hee</uri> </author> <author> <name>&quot;Samei, Ebrahim&quot;</name> <uri>https://zbmath.org/authors/?q=ai:samei.ebrahim</uri> </author> <author> <name>&quot;Wiersma, Matthew&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wiersma.matthew</uri> </author> <content type="text">Summary: Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors \((A,B)\mapsto A\otimes_{\alpha } B\), where \(A\otimes_\alpha B\) is a cross norm completion of \(A\odot B\) for each pair of \(\mathrm{C}^\ast\)-algebras \(A\) and \(B\). For the first class of bifunctors considered \((A,B)\mapsto A{\otimes_p} B\) (\(1\leq p\leq \infty\)), \(A{\otimes_p} B\) is a Banach algebra cross-norm completion of \(A\odot B\) constructed in a fashion similar to \(p\)-pseudofunctions \(\mathrm{PF}^*_p(G)\) of a locally compact group. Taking a cue from the recently introduced symmetrized \(p\)-pseudofunctions due to \textit{B.-B. Liao} and \textit{G.-L. Yu} [``K-theory of group Banach algebras and Banach property RD'', Preprint (2017), \url{arXiv:1708.01982}] and later by the second and the third named authors [\textit{E.~Samei} and \textit{M.~Wiersma}, J. Funct. Anal. 286, No.~2, Article ID 110228, 32~p. (2024; Zbl 1537.46042)], we also consider \({\otimes_{p,q}}\) for H枚lder conjugate \(p,q\in [1,\infty ]\) -- a Banach \(*\)-algebra analogue of the tensor product \({\otimes_{p,q}}\). By taking enveloping \(\mathrm{C}^\ast\)-algebras of \(A{\otimes_{p,q}} B\), we arrive at a third bifunctor \((A,B)\mapsto A{\otimes_{\mathrm{C}^\ast_{p,q}}} B\) where the resulting algebra \(A{\otimes_{\text C^*_{p,q}}} B\) is a \(\mathrm{C}^\ast\)-algebra. For \(G_1\) and \(G_2\) belonging to a large class of discrete groups, we show that the tensor products \(\mathrm{C}^\ast_{\text r}(G_1){\otimes_{\mathrm{C}^\ast_{p,q}}}\mathrm{C}^\ast_{\text r}(G_2)\) coincide with a Brown-Guentner type \(\mathrm{C}^\ast\)-completion of \(\ell^1(G_1\times G_2)\) and conclude that if \(2\leq p'&lt;p\leq \infty \), then the canonical quotient map \(\mathrm{C}^\ast_{\text r}(G){\otimes_{\mathrm{C}^\ast_{p,q}}}\mathrm{C}^\ast_{\text r}(G)\to \mathrm{C}^\ast_{\text r}(G){\otimes_{\mathrm{C}^\ast_{p,q}}}\mathrm{C}^\ast_{\text r}(G)\) is not injective for a large class of non-amenable discrete groups possessing both the rapid decay property and Haagerup's approximation property. A Banach \(*\)-algebra \(A\) is symmetric if the spectrum \(\mathrm{Sp}_A(a^*a)\) is contained in \([0,\infty )\) for every \(a\in A\), and rigidly symmetric if \(A\otimes_{\gamma } B\) is symmetric for every \(\mathrm{C}^\ast\)-algebra \(B\). A theorem of Kugler [\textit{W.~Kugler}, Math. Z. 168, 241--262 (1979; Zbl 0394.43004)] asserts that every type I \(\mathrm{C}^\ast\)-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kugler's theorem by showing for \(\mathrm{C}^\ast\)-algebras \(A\) and \(B\) that \(A\otimes_{\gamma }B\) is symmetric if and only if \(A\) or \(B\) is type I. In particular, a \(\mathrm{C}^\ast\)-algebra is rigidly symmetric if and only if it is type~I. This strongly settles a question of \textit{H.~Leptin} and \textit{D.~Poguntke} from [J. Funct. Anal. 33, 119--134 (1979; Zbl 0414.43004)] and corrects an error in the literature.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">An operator-valued free Poincar茅 inequality</title> <id>https://zbmath.org/1552.46042</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.46042" /> <author> <name>&quot;Ito, Hyuga&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ito.hyuga</uri> </author> <content type="text">Summary: The purpose of this short note is to give an operator-valued free Poincar茅 inequality, which provides a simple proof to (an improvement of) a lemma of \textit{D.~Voiculescu} [Int. Math. Res. Not. 2000, No.~2, 79--106 (2000; Zbl 0952.46038)] asserting that the kernel of the free difference quotient is exactly the coefficients.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Curve shortening flow on Riemann surfaces with conical singularities</title> <id>https://zbmath.org/1552.53089</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.53089" /> <author> <name>&quot;Roidos, Nikolaos&quot;</name> <uri>https://zbmath.org/authors/?q=ai:roidos.nikolaos</uri> </author> <author> <name>&quot;Savas-Halilaj, Andreas&quot;</name> <uri>https://zbmath.org/authors/?q=ai:savas-halilaj.andreas</uri> </author> <content type="text">The results in this article revolve around the question of curve shortening in the setting of Riemann surfaces that exhibit conformal conical singularities. The authors reduce this study to the investigation of degenerate quasilinear parabolic equations for which they discuss short time existence, uniqueness, and regularity of the flow. The approach implements the Gauss-Bonnet formula for singular surfaces with boundary, properties of sectorial operators and various tools in Mellin-Sobolev spaces. Reviewer: Marius Ghergu (Dublin)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/47B"> <title type="text">Oversampling on a class of symmetric regular de Branges spaces</title> <id>https://zbmath.org/1552.94017</id> <updated>2025-03-24T18:23:18.698908Z</updated> <link href="https://zbmath.org/1552.94017" /> <author> <name>&quot;Silva, Luis O.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:silva.luis-o</uri> </author> <author> <name>&quot;Toloza, Julio H.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:toloza.julio-h</uri> </author> <content type="text">Summary: A de Branges space \(\mathcal{B}\) is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map \(F(z) \mapsto F(-z)\). Let \(K_\mathcal{B} (z,w)\) be the reproducing kernel in \(\mathcal{B}\) and \(S_\mathcal{B}\) be the operator of multiplication by the independent variable with maximal domain in \(\mathcal{B}\). Loosely speaking, we say that \(\mathcal{B}\) has the \(\ell_p\)-oversampling property relative to a proper subspace \(\mathcal{A}\) of it, with \(p \in (2, \infty]\), if there exists \(J_{\mathcal{A} \mathcal{B}} : \mathbb{C} \times \mathbb{C} \to \mathbb{C}\) such that \(J(\cdot,w) \in \mathcal{B}\) for all \(w \in \mathbb{C}\), \[ \begin{aligned} &amp;\sum_{\lambda \in \sigma(S_{\mathcal{B}}^{\gamma})} \left(\frac{\left\vert J_{\mathcal{A}\mathcal{B}}(z,\lambda) \right\vert}{K_\mathcal{B}(\lambda,\lambda)^{1/2}}\right)^{p/(p-1)} &lt;\infty \quad \text{and} \\ &amp;F(z) = \sum_{\lambda \in \sigma(S_{\mathcal{B}}^{\gamma})} \frac{J_{\mathcal{A}\mathcal{B}}(z,\lambda)}{K_\mathcal{B}(\lambda,\lambda)}F(\lambda), \end{aligned} \] for all \(F \in \mathcal{A}\) and almost every self-adjoint extension \(S_{\mathcal{B}}^{\gamma }\) of \(S_\mathcal{B}\). This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper, we provide sufficient conditions for a symmetric, regular de Branges space to have the \(\ell_p\)-oversampling property relative to a chain of de Branges subspaces of it.</content> </entry> </feed>