Recent zbMATH articles in MSC 46https://zbmath.org/atom/cc/462024-03-13T18:33:02.981707ZUnknown authorWerkzeugOracle computability of conditional expectations onto subfactorshttps://zbmath.org/1528.031582024-03-13T18:33:02.981707Z"Goldbring, Isaac"https://zbmath.org/authors/?q=ai:goldbring.isaacSummary: We initiate the effective study of conditional expectations onto subfactors. Our main result is that if \(M\) is an existentially closed II\(_1\) factor with a w-spectral gap subfactor \(N\), then the conditional expectation function onto \(N\) can be computed from a (Turing) oracle that computes a presentation of \(M\), the inclusion of \(N\) into \(M\), and a spectral gap function for the pair \((M,N)\).McShane-Whitney extensions in constructive analysishttps://zbmath.org/1528.032492024-03-13T18:33:02.981707Z"Petrakis, Iosif"https://zbmath.org/authors/?q=ai:petrakis.iosifSummary: Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued functions on a totally bounded space are uniformly dense in the set of uniformly continuous functions. Through the introduced notion of a McShane-Whitney pair we describe the constructive content of the original McShane-Whitney extension and examine how the properties of a Lipschitz function defined on the subspace of the pair extend to its McShane-Whitney extensions on the space of the pair. Similar McShane-Whitney pairs and extensions are established for Hölder functions and \(\nu\)-continuous functions, where \(\nu\) is a modulus of continuity. A Lipschitz version of a fundamental corollary of the Hahn-Banach theorem, and the approximate McShane-Whitney theorem are shown.\(L\)-functions of noncommutative torihttps://zbmath.org/1528.110502024-03-13T18:33:02.981707Z"Nikolaev, Igor V."https://zbmath.org/authors/?q=ai:nikolaev.igor-vladimirovich|nikolaev.igor-vasilievichSummary: We introduce an analog of the \(L\)-function for noncommutative tori. It is proved that such a function coincides with the Hasse-Weil \(L\)-function of an elliptic curve with complex multiplication. As a corollary, one gets a localization formula for the noncommutative tori with real multiplication.Orthant-strictly monotonic norms, generalized top-\(k\) and \(k\)-support norms and the \(\ell_0\) pseudonormhttps://zbmath.org/1528.150192024-03-13T18:33:02.981707Z"Chancelier, Jean-Philippe"https://zbmath.org/authors/?q=ai:chancelier.jean-philippe"De Lara, Michel"https://zbmath.org/authors/?q=ai:de-lara.michelSummary: The so-called \(\ell_0\) pseudonorm on the Euclidean space \(\mathbb{R}^d\) counts the number of nonzero components of a vector. We say that a sequence of norms is strictly increasingly graded (with respect to the \(\ell_0\) pseudonorm) if it is nondecreasing and that the sequence of norms of a vector \(x\) becomes stationary exactly at the index \(\ell_0(x)\). In this paper, with any (source) norm, we associate sequences of generalized top-\(k\) and \(k\)-support norms, and we also introduce the new class of orthant-strictly monotonic norms (that encompasses the \(\ell_p\) norms, but for the extreme ones). Then, we show that an orthant-strictly monotonic source norm generates a sequence of generalized top-\(k\) norms which is strictly increasingly graded. With this, we provide a systematic way to generate sequences of norms with which the level sets of the \(\ell_0\) pseudonorm are expressed by means of the difference of two norms. Our results rely on the study of orthant-strictly monotonic norms.Representing structured semigroups on étale groupoid bundleshttps://zbmath.org/1528.201032024-03-13T18:33:02.981707Z"Bice, Tristan"https://zbmath.org/authors/?q=ai:bice.tristan-matthewSummary: We examine a semigroup analogue of the Kumjian-Renault representation of \(C^*\)-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as `slice-sections' of groupoid bundles.Decay estimates for matrix coefficients of unitary representations of semisimple Lie groupshttps://zbmath.org/1528.220072024-03-13T18:33:02.981707Z"Cowling, Michael G."https://zbmath.org/authors/?q=ai:cowling.michael-gSummary: Let \(G\) be a connected semisimple Lie group with finite centre and \(K\) be a maximal compact subgroup thereof. Given a function \(u\) on \(G\), we define \(\mathcal{A} u\) to be the root-mean-square average over \(K\), acting both on the left and the right, of \(u\). We take a positive-real-valued spherical function \(\phi_\lambda\) on \(G\), and study the Banach convolution algebra of \(\mathsf{C}_c (G)\)-functions \(u\) with the norm \(\| u \|_{(\lambda)} : = \int_G \mathcal{A} u(x) \phi_\lambda (x) dx\). The \(C^\ast\) completion of this algebra is an exotic \(C^\ast\)-algebra on \(G\), in the sense that it lies ``between'' the reduced \(C^\ast\)-algebra of \(G\) and the full \(C^\ast\)-algebra of \(G\), and in the sense that it arise as the completion of a star-algebra that does not contain an approximate identity.
Using functional analysis and representation theory, we show that for all unitary representations \(\pi\) of \(G\), there exists a unique minimal positive-real-valued spherical function \(\phi_\lambda\) on \(G\) such that \(\mathcal{A} \langle \pi (\cdot) \xi, \eta \rangle \leq \| \xi \|_{\mathcal{H}_\pi} \| \eta \|_{\mathcal{H}_\pi} \phi_\lambda\). This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates \(| \langle \pi (\cdot) \xi, \eta \rangle | \leq C(\xi, \eta) \phi_\lambda\) for \(K\)-finite or smooth vectors \(\xi\) and \(\eta\), and it exhibits different decay rates in different directions at infinity in \(G\). Further, if we assume the latter inequality with arbitrary \(C(\xi, \eta)\), we can prove the former inequality and then return to the latter inequality with explicit knowledge of \(C(\xi, \eta)\). Finally, it holds everywhere in \(G\), in contrast to asymptotic estimates which are not global and to \(\mathsf{L}^p\) estimates which carry no pointwise information.Vanishing of certain equivariant distributions on spherical spaces for quasi-split groupshttps://zbmath.org/1528.220082024-03-13T18:33:02.981707Z"Lu, Hengfei"https://zbmath.org/authors/?q=ai:lu.hengfeiThis paper contains an important result concerning the vanishing of some eigendistributions on a quasi-split real reductive group. Let \(G\) be a quasi-split real reductive group, \(B\) be a standard Borel subgroup of \(G\), \(U\) be the corresponding unipotent radical of \(B\), and let \(\Psi\) be a non-degenerate character of \(U\). Let \(H\) be a spherical subgroup of \(G\), and let \(\chi\) be a character of \(H\). Let \(Z\) be the complement to the union of open \(B\times H\)-double cosets in \(G\), and let \(\mathfrak{z}\) be the center of the enveloping algebra \(\mathscr{U}(\mathfrak{g})\) of the Lie algebra \(\mathfrak{g}\) of \(G\). The author proves (Theorem 1.1) that there are no non-zero \(\mathfrak{z}\)-eigen \((U\times H\), \(\psi \times \chi\))-equivariant distributions supported on \(Z\).
Theorem 1.1 is a generalization of a theorem from [\textit{A. Aizenbud} and \textit{D. Gourevitch}, Math. Z. 279, No. 3--4, 745--751 (2015; Zbl 1315.22014)], where the authors obtained a similar result for a split real reductive group \(G\).
Reviewer: Allan Merino (Boston)On real-valued homogeneous polynomials with many variableshttps://zbmath.org/1528.260202024-03-13T18:33:02.981707Z"Raposo, Anselmo jun."https://zbmath.org/authors/?q=ai:raposo.anselmo-jun"Teixeira, Katiuscia B."https://zbmath.org/authors/?q=ai:teixeira.katiuscia-bLet \(P_m\) be a real-valued \(m\)-homogeneous polynomial. Denote by
\[
\Vert P_m \Vert :=\sup_{\Vert (x_1,x_2,\ldots, x_n)\Vert _{\infty}\leq 1} |P_m(x_1,x_2,\ldots, x_n)|,
\]
and by \(|P_m|_p\) the \(\ell_p\) norm of the coefficients of \(P_m\). The classical Bohnenblust-Hille inequality says that for each \(p\geq \frac{2m}{m+1}\) there exists a constant \(C_{p,m} \geq 1,\) independent of the number of variables \(n,\) such that \(|P_m|_p \leq C_{p,m}\Vert P_m \Vert.\) The value \(\frac{2m}{m+1}\) for \(p\) is the smallest possible. When \(p=\frac{2m}{m+1}\) we will denote by \( C_m:=C_{m,p}.\) In the present paper for all \(p \in (0, \infty]\) and any positive integer \(m\), the authors investigate the asymptotic behavior of \(\sup\{|P_m|_p: \ \Vert P_m \Vert=1\}.\) They prove that for all \(p \in [2, \infty],\) the following holds \(\lim_{m \to \infty} \left( \sup \left\{ |P_m|_p: \ \Vert P_m \Vert =1 \right\}\right)^{1/m}=2.\) When \(p\in (0,2)\) the limit is infinity. Moreover, the authors show that
\[
\lim_{m \to \infty} \left( \sup \left \{ |P_m|_{\frac{2m}{m+1 }}: \ \Vert P_m \Vert =1 \right\}\right)^{1/m}=2.
\]
Hence, \(\lim_{m \to \infty} C_m^{1/m}=2,\) independently of dimension.
Reviewer: Olga M. Katkova (Boston)Uniform Poincaré inequality in o-minimal structureshttps://zbmath.org/1528.260252024-03-13T18:33:02.981707Z"Valette, Anna"https://zbmath.org/authors/?q=ai:valette-stasica.anna"Valette, Guillaume"https://zbmath.org/authors/?q=ai:valette.guillaumeSummary: Wefirst define the trace on a domain \(\Omega\) which is definable in an o-minimal structure. We then show that every function \(u \in W^{1,p}(\Omega)\) vanishing on the boundary in the trace sense satisfies Poincaré inequality. We finally show, given a definable family of domains \((\Omega_t)_{t \in\mathbb{R}^k}\), that the constant of this inequality remains bounded, if so does the volume of \(\Omega_t\).Hardy-Littlewood-Sobolev inequality and existence of the extremal functions with extended kernelhttps://zbmath.org/1528.260302024-03-13T18:33:02.981707Z"Liu, Zhao"https://zbmath.org/authors/?q=ai:liu.zhao.1|liu.zhaoSummary: In this paper, we consider the following Hardy-Littlewood-Sobolev inequality with extended kernel
\[
\int_{\mathbb{R}_+^n}\int_{\partial\mathbb{R}^n_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) \mathrm{d}y\mathrm{d}x\leqslant C_{n,\alpha,\beta,p} \| f\|_{L^p (\partial\mathbb{R}_+^n)} \| g\|_{L^{q'}(\mathbb{R}_+^n)},
\tag{0.1}
\]
for any nonnegative functions \(f\in L^p (\partial \mathbb{R}_+^n)\), \(g\in L^{q'}(\mathbb{R}_+^n)\) and \(p,\, q'\in (1,\,\infty )\), \(\beta \geqslant 0\), \(\alpha +\beta >1\) such that \(\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}-\frac{\alpha +\beta -1}{n}=1\).
We prove the existence of all extremal functions for (0.1). We show that if \(f\) and \(g\) are extremal functions for (0.1) then both of \(f\) and \(g\) are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler-Lagrange equations by using Pohozaev identity.Some porosity-type properties of sets related to the \(d\)-Hausdorff contenthttps://zbmath.org/1528.280032024-03-13T18:33:02.981707Z"Tyulenev, A. I."https://zbmath.org/authors/?q=ai:tyulenev.alexander-iSummary: Let \(S\subset\mathbb{R}^n\) be a nonempty set. Given \(d\in [0,n)\) and a cube \(\overline{Q}\subset\mathbb{R}^n\) with side length \(l=l(\overline{Q}) \in (0,1]\), we show that if the \(d\)-Hausdorff content \(\mathcal{H}^d_{\infty}(\overline{Q}\cap S)\) of the set \(\overline{Q}\cap S\) satisfies the inequality \(\mathcal{H}^d_{\infty}(\overline{Q}\cap S)<\overline{\lambda}l^d\) for some \(\overline{\lambda}\in (0,1)\), then the set \(\overline{Q}\setminus S\) contains a specific cavity. More precisely, we prove the existence of a pseudometric \(\rho=\rho_{S,d}\) such that for every sufficiently small \(\delta>0\) the \(\delta \)-neighborhood \(U^\rho_{\delta}(S)\) of \(S\) in the pseudometric \(\rho\) does not cover \(\overline{Q} \). Moreover, we establish the existence of constants \(\overline{\delta}=\overline{\delta}(n,d,\overline{\lambda})>0\) and \(\underline{\gamma}=\underline{\gamma}(n,d,\overline{\lambda})>0\) such that \(\mathcal L^n(\overline{Q}\setminus U^{\rho}_{\delta l}(S)) \geq \underline{\gamma} l^n\) for all \(\delta\in (0,\overline{\delta})\), where \(\mathcal L^n\) is the Lebesgue measure. If in addition the set \(S\) is lower content \(d\)-regular, we prove the existence of a constant \(\underline{\tau}=\underline{\tau}(n,d,\overline{\lambda})>0\) such that the cube \(\overline{Q}\) is \(\underline{\tau} \)-porous. The sharpness of the results is illustrated by several examples.Extensions and traces of BV functions in rough domains and generalized Cheeger setshttps://zbmath.org/1528.280092024-03-13T18:33:02.981707Z"Gui, Changfeng"https://zbmath.org/authors/?q=ai:gui.changfeng"Hu, Yeyao"https://zbmath.org/authors/?q=ai:hu.yeyao"Li, Qinfeng"https://zbmath.org/authors/?q=ai:li.qinfengSummary: Via the method of interior approximation, we prove trace and extension results for BV functions defined on a bounded domain \(\Omega\subset\mathbb{R}^n\) such that \(\Omega\) satisfies
\[
\mathscr{H}^{n-1}(\partial\Omega\setminus\Omega^0) < \infty, \tag{0.1}
\]
where \(\mathscr{H}^{n-1}\) is the \((n-1)\) dimensional Hausdorff measure and \(\Omega^0\) is the measure-theoretic exterior of \(\Omega\). Stronger results are obtained on a subclass of such domains, which are outward minimizing domains. We also obtain new weak regularity results for generalized Cheeger sets as byproducts.Approximation of functions on rays in \(\mathbb{R}^n\) by solutions to convolution equationshttps://zbmath.org/1528.300132024-03-13T18:33:02.981707Z"Volchkov, V. V."https://zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: This is a first study of approximation of continuous functions on rays in \(\mathbb{R}^n\) by smooth solutions to a multidimensional convolution equation with a radial convolutor. We obtain an analog of the well-known Carleman's Theorem on tangent approximation by entire functions. As consequences, we give some new results of interest for the theory of convolution equations. These results concern the density in \(\mathbb{C}\) of the range of some solutions to the convolution equation as well as the possible growth of solutions on rays in \(\mathbb{R}^n \).Coefficient problems of quasi-convex mappings of type B on the unit ball in complex Banach spaceshttps://zbmath.org/1528.320262024-03-13T18:33:02.981707Z"Zhang, Ruyu"https://zbmath.org/authors/?q=ai:zhang.ruyu"Ouyang, Dongling"https://zbmath.org/authors/?q=ai:ouyang.dongling"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpengSummary: In this paper, the sharp solutions of Fekete-Szegö problems are provided for class of quasi-convex mappings \(f_1\) of type B and class of quasi-convex mappings \(f_2\) of type B and order \(\alpha\) defined on the unit ball in a complex Banach space, respectively, where \(x=0\) is a zero of order \(k+1\) of \(f_i(x)-x(i=1,2)\). Compare with some recent works, our main theorems hold without additional restrictive conditions. Also, the proof of our main theorems are more simple than those given in the previous results.Entropy solutions of nonlinear \(p(x)\)-parabolic inequalitieshttps://zbmath.org/1528.350722024-03-13T18:33:02.981707Z"Akdim, Youssef"https://zbmath.org/authors/?q=ai:akdim.youssef"Chakir, Allalou"https://zbmath.org/authors/?q=ai:allalou.chakir"El gorch, Nezha"https://zbmath.org/authors/?q=ai:el-gorch.nezha"Mekkour, Mounir"https://zbmath.org/authors/?q=ai:mekkour.mounirSummary: In this paper we prove the existence of entropy solutions for weighted \(p(x)\)-parabolic problem associated with the equation:
\[
\frac{\partial u}{\partial t} + Au 0 g(u)\omega(x) \vert \nabla u\vert^{p(x)} + f \quad\text{in }\Omega\times (0,T),
\]
where the operator \(Au = -\operatorname{div}\left(\omega(x) \vert \nabla u\vert^{p(x) - 2} \nabla u\right)\) and on the right-hand side \(f\) belongs to \(L^1(\Omega\times (0,T))\) and \(\omega(x)\) is a weight function.Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent \(\gamma > 3\)https://zbmath.org/1528.350982024-03-13T18:33:02.981707Z"Oschmann, Florian"https://zbmath.org/authors/?q=ai:oschmann.florian"Pokorný, Milan"https://zbmath.org/authors/?q=ai:pokorny.milan.1|pokorny.milanSummary: We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent \(\gamma > 3\) instead of the known value \(\gamma > 6\). The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.Global well-posedness of the Navier-Stokes equations in homogeneous Besov spaces on the half-spacehttps://zbmath.org/1528.351012024-03-13T18:33:02.981707Z"Watanabe, Keiichi"https://zbmath.org/authors/?q=ai:watanabe.keiichi.1Summary: Consider the Stokes equations in the half-space \(\mathbb{R}_+^n\), \(n \geqq 2\). It is shown that the negative of the Stokes operator defined on the homogeneous Besov space \(\dot{B}_{p, q, \sigma}^s (\mathbb{R}_+^n)\) generates a bounded strongly continuous semigroup in \(\dot{B}_{p, q}^s (\mathbb{R}_+^n)\) provided that \(1 < p < \infty\), \(1 \leqq q < \infty\), and \(- 1 + 1 / p < s < 1 / p\). As a by-product, the maximal \(L^q\)-regularity of the Stokes operator is obtained, admitting the limiting case \(q = 1\). This will be applied to develop the global well-posedness result for the incompressible Navier-Stokes equations in the maximal \(L^1\)-regularity class provided that the initial data are small in \(\dot{B}_{p, 1}^{- 1 + n / p} (\mathbb{R}_+^n)\) with \(n - 1 < p < \infty\). The dependence of the solution on initial data and the large time behavior of the solution are investigated.Ground states for the NLS equation with combined local nonlinearities on noncompact metric graphshttps://zbmath.org/1528.351622024-03-13T18:33:02.981707Z"Li, Xiaoguang"https://zbmath.org/authors/?q=ai:li.xiaoguang"Zhang, Guoqing"https://zbmath.org/authors/?q=ai:zhang.guoqing"Liu, Lele"https://zbmath.org/authors/?q=ai:liu.leleSummary: We investigate the existence of ground states for the nonlinear Schrödinger equation with combined local \(L^2\)-critical and subcritical nonlinearities on noncompact metric graphs. The interplay of the combined local nonlinearities creates some new phenomena. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on the topological and metric properties of the noncompact metric graphs. Our results rely on the Concentration-compactness principle and an application of Gagliardo-Nirenberg inequalities.Strong instability of standing waves for the divergence Schrödinger equation with inhomogeneous nonlinearityhttps://zbmath.org/1528.351732024-03-13T18:33:02.981707Z"Zheng, Bowen"https://zbmath.org/authors/?q=ai:zheng.bowen"Zhu, Wenjing"https://zbmath.org/authors/?q=ai:zhu.wenjingSummary: This paper considers a class of Schrödinger type equations with a divergence dispersive term and inhomogeneous nonlinearity. We first establish the variational characterization of ground states by introducing a weighted Sobolev embedding theorem. Then, based on a localized variance-type estimate, we prove that the standing waves are strongly unstable using blow-up.Existence of solutions and their behavior for the anisotropic quasi-geostrophic equation in Sobolev and Sobolev-Gevrey spaceshttps://zbmath.org/1528.352082024-03-13T18:33:02.981707Z"Melo, Wilberclay G."https://zbmath.org/authors/?q=ai:melo.wilberclay-g"Santos, Thyago S. R."https://zbmath.org/authors/?q=ai:santos.thyago-s-r"dos Santos Costa, Natielle"https://zbmath.org/authors/?q=ai:dos-santos-costa.natielleSummary: This paper establishes that the anisotropic quasi-geostrophic equation, with orders of fractional dissipation \(\alpha\) and \(\beta\), admits a unique mild solution \(\theta\) in homogeneous Sobolev and Sobolev-Gevrey spaces \(\dot{H}_{a, \sigma}^s (\mathbb{R}^2)\) (with \(a \geq 0\), \(\sigma \geq 1\), \(\alpha \in (\frac{1}{2}, \frac{3}{4}]\), \(\beta \in (\frac{1}{2}, \frac{3}{4}]\) and \(\max \{2 \alpha - 1, 2 \beta - 1\} \leq s < 1 - |\alpha - \beta|\)) provided that the initial data \(\theta_0\) is small enough in these spaces (the critical cases \(\alpha = \frac{1}{2}\) and \(\beta = \frac{1}{2}\) are studied as well). Moreover, this work also studies the behavior of this same solution \(\theta\) through the following decay rate:
\[
\limsup_{t \to \infty} t^{\frac{\kappa}{2 \max \{\alpha, \beta\}}} \|\theta (t)\|_{\dot{H}_{a, \sigma}^\kappa} = 0,
\]
for all \(\kappa \geq 0\), where \(a \geq 0\), \(\sigma \geq 1\), \(\alpha \in (\frac{1}{2}, \frac{3}{4}]\), \(\beta \in (\frac{1}{2}, \frac{3}{4}]\) and \(\max \{2 \alpha - 1, 2 \beta - 1\} \leq s \leq \min \{2 - 2 \alpha, 2 - 2 \beta\})\). It is important to emphasize that the limit superior above is a consequence of the Gevrey regularity of \(\theta\) and of the fact that
\[
\lim_{t \to \infty} \|\theta (t)\|_{L^2} = 0,
\]
if it is assumed that \(\theta_0 \in L^2 (\mathbb{R}^2)\).\(\lambda\)-statistical convergence of interval numbers of order \(\alpha\)https://zbmath.org/1528.400042024-03-13T18:33:02.981707Z"Esi, Ayhan"https://zbmath.org/authors/?q=ai:esi.ayhan"Esi, Ayten"https://zbmath.org/authors/?q=ai:esi.aytenStatistical convergence, which is a generalization of classical convergence, has been introduced by \textit{H. Fast} [Colloq. Math. 2, 241--244 (1951; Zbl 0044.33605)], and \textit{I. J. Schoenberg} [Am. Math. Mon. 66, 361--375, 562--563 (1959; Zbl 0089.04002)] has studied this concept from the perspective of summability. These studies dealt with real or complex sequences, but several authors extended this idea to fuzzy numbers. Besides this, interval arithmetic was introduced by \textit{P. S. Dwyer} [Linear computations. New York: John Wiley \& Sons, Inc (1951; Zbl 0044.12804)] and it has been well studied since it has noteworthy value as a computational device.
In the present paper, the authors introduce some of the basic concepts of summability theory for a sequence of interval numbers such as strong \(\lambda\)-summability of order \(\alpha\), \(\lambda\)-statistical convergence of order \(\alpha\), where \(\lambda=(\lambda_n)\) is a nondecreasing sequence of positive numbers such that \(\lambda_{n+1} \leq \lambda_n+1 ,\) \(\lambda_1=1,\) and \(\lambda_n \rightarrow \infty\) as \(n \rightarrow \infty.\) The basic properties of these concepts are obtained and inclusions between the spaces of \(\lambda\)-statistical convergent sequences of different orders are provided. Also, some interval valued sequence spaces are investigated in terms of being monotone, convergence free, solid and complete.
For the entire collection see [Zbl 1495.40002].
Reviewer: Tuğba Yurdakadim (Bilecik)Polynomial approximations in a generalized Nyman-Beurling criterionhttps://zbmath.org/1528.410472024-03-13T18:33:02.981707Z"Alouges, François"https://zbmath.org/authors/?q=ai:alouges.francois"Darses, Sébastien"https://zbmath.org/authors/?q=ai:darses.sebastien"Hillion, Erwan"https://zbmath.org/authors/?q=ai:hillion.erwanSummary: The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on \((0, \infty)\), involving dilations of the fractional part function by factors \(\theta_k\in(0, 1)\), \(k \geq 1\). Randomizing the \(\theta_k\) gener
By the way, I had editing rights over that page, maybe they should be
transferred to you. The person I was in touch with was '''Robert
Gieseke''' <>ates new structures and criteria. One of them is a sufficient condition for RH that splits into
\begin{itemize}
\item[(i)] showing that the indicator function can be approximated by convolution with the fractional part,
\item[(ii)] a control on the coefficients of the approximation.
\end{itemize}
This self-contained paper generalizes conditions (i) and (ii) that involve a \(\sigma_0\in(1/2, 1)\), and imply \(\zeta (\sigma +it) \neq 0\) in the strip \(1/2 < \sigma \leq \sigma_0 < 1\). We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.A Chebyshev-type alternation theorem for best approximation by a sum of two algebrashttps://zbmath.org/1528.410482024-03-13T18:33:02.981707Z"Asgarova, Aida Kh."https://zbmath.org/authors/?q=ai:asgarova.aida-kh"Huseynli, Ali A."https://zbmath.org/authors/?q=ai:huseynli.ali-a"Ismailov, Vugar E."https://zbmath.org/authors/?q=ai:ismailov.vugar-eSummary: Let \(X\) be a compact metric space, \(C(X)\) be the space of continuous real-valued functions on \(X\) and \(A_1,A_2\) be two closed subalgebras of \(C(X)\) containing constant functions. We consider the problem of approximation of a function \(f\in C(X)\) by elements from \(A_1+A_2\). We prove a Chebyshev-type alternation theorem for a function \(u_0 \in A_1+A_2\) to be a best approximation to \(f\).\(n\)-best kernel approximation in reproducing kernel Hilbert spaceshttps://zbmath.org/1528.410782024-03-13T18:33:02.981707Z"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.taoThe paper considers a class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc. The existence of \(n\)-best kernel approximation under the norm of such a reproducing kernel Hilbert space is investigated. Assuming that the reproducing kernel satisfies an analyticity condition, an infinite-norm property, and a uniform boundedness condition, the existence is proved. Applications to the classical Hardy space, weighted Hardy spaces, and stochastic \(n\)-best approximation are also discussed.
Reviewer: Haizhang Zhang (Guangzhou)Elton's near unconditionality of bases as a threshold-free form of greedinesshttps://zbmath.org/1528.410882024-03-13T18:33:02.981707Z"Albiac, Fernando"https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José L."https://zbmath.org/authors/?q=ai:ansorena.jose-luis"Berasategui, Miguel"https://zbmath.org/authors/?q=ai:berasategui.miguelSummary: Elton's near unconditionality and quasi-greediness for largest coefficients are two properties of bases that made their appearance in functional analysis from very different areas of research. One of our aims in this note is to show that, oddly enough, they are connected to the extent that they are equivalent notions. We take advantage of this new description of the former property to further the study of the threshold function associated with near unconditionality. Finally, we make a contribution to the isometric theory of greedy bases by characterizing those bases that are 1-quasi-greedy for largest coefficients.Sparse approximation using new greedy-like bases in superreflexive spaceshttps://zbmath.org/1528.410892024-03-13T18:33:02.981707Z"Albiac, Fernando"https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José L."https://zbmath.org/authors/?q=ai:ansorena.jose-luis"Berasategui, Miguel"https://zbmath.org/authors/?q=ai:berasategui.miguelAuthors' abstract: This paper is devoted to theoretical aspects of optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of non-linear \(m\)-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regard to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original Dilworth-Kalton-Kutzarova method [\textit{S. J. Dilworth} et al., Stud. Math. 159, No. 1, 67--101 (2003; Zbl 1056.46014)] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.
Reviewer: Francisco Pérez Acosta (La Laguna)Bidemocratic bases and their connections with other greedy-type baseshttps://zbmath.org/1528.410902024-03-13T18:33:02.981707Z"Albiac, Fernando"https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José L."https://zbmath.org/authors/?q=ai:ansorena.jose-luis"Berasategui, Miguel"https://zbmath.org/authors/?q=ai:berasategui.miguel"Berná, Pablo M."https://zbmath.org/authors/?q=ai:berna.pablo-manuel"Lassalle, Silvia"https://zbmath.org/authors/?q=ai:lassalle.silviaSummary: In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each \(1<p<\infty\) the space \(\ell_p\) has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially democratic bases.New parameters and Lebesgue-type estimates in greedy approximationhttps://zbmath.org/1528.410912024-03-13T18:33:02.981707Z"Albiac, Fernando"https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José L."https://zbmath.org/authors/?q=ai:ansorena.jose-luis"Berná, Pablo M."https://zbmath.org/authors/?q=ai:berna.pablo-manuelSummary: The purpose of this paper is to quantify the size of the Lebesgue constants \((\boldsymbol{L}_m)_{m=1}^\infty\) associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined \textit{linearly} with the sequence of unconditionality parameters \((\boldsymbol{k}_m)_{m=1}^\infty\) determines the growth of \((\boldsymbol{L}_m)_{m=1}^\infty\). Multiple theoretical applications and computational examples complement our study.Extensions and new characterizations of some greedy-type baseshttps://zbmath.org/1528.410922024-03-13T18:33:02.981707Z"Berasategui, Miguel"https://zbmath.org/authors/?q=ai:berasategui.miguel"Berná, Pablo M."https://zbmath.org/authors/?q=ai:berna.pablo-manuel"Chu, Hùng Việt"https://zbmath.org/authors/?q=ai:chu.hung-viet.1Summary: Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich baseshttps://zbmath.org/1528.410932024-03-13T18:33:02.981707Z"Berasategui, Miguel"https://zbmath.org/authors/?q=ai:berasategui.miguel"Lassalle, Silvia"https://zbmath.org/authors/?q=ai:lassalle.silviaSummary: We introduce and study the notion of weak semi-greedy systems -- which is inspired in the concepts of semi-greedy and branch semi-greedy systems and weak thresholding sets-, and prove that in infinite dimensional Banach spaces, the notions of \textit{semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from [\textit{P. M. Berná}, J. Math. Anal. Appl. 470, No. 1, 218--225 (2019; Zbl 1457.46015); \textit{S. J. Dilworth} et al., Stud. Math. 159, No. 1, 67--101 (2003; Zbl 1056.46014); \textit{S. J. Dilworth} et al., J. Funct. Anal. 263, No. 12, 3900--3921 (2012; Zbl 1267.46026)]. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.Performance of the thresholding greedy algorithm with larger greedy sumshttps://zbmath.org/1528.410942024-03-13T18:33:02.981707Z"Chu, Hùng Việt"https://zbmath.org/authors/?q=ai:chu.hung-viet.1Summary: The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor \(\lambda \geqslant 1\). We introduce the so-called \(\lambda\)-almost greedy and \(\lambda\)-partially greedy bases. The case when \(\lambda = 1\) gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is \(\lambda\)-almost greedy for all (some) \(\lambda \geqslant 1\). However, for each \(\lambda > 1\), there exists an unconditional basis that is \(\lambda\)-partially greedy but is not 1-partially greedy. Furthermore, we investigate and give examples when a basis is
\begin{itemize}
\item[(1)] not almost greedy with constant 1 but is \(\lambda\)-almost greedy with constant 1 for some \(\lambda > 1\), and
\item[(2)] not strong partially greedy with constant 1 but is \(\lambda\)-partially greedy with constant 1 for some \(\lambda > 1\).
\end{itemize}
Finally, we prove various characterizations of different greedy-type bases.A simple proof of the Grünbaum conjecturehttps://zbmath.org/1528.410962024-03-13T18:33:02.981707Z"Derȩgowska, Beata"https://zbmath.org/authors/?q=ai:deregowska.beata"Lewandowska, Barbara"https://zbmath.org/authors/?q=ai:lewandowska.barbaraSummary: Let \(\lambda_{\mathbb{K}}(m)\) denote the maximal absolute projection constant over the subspaces of dimension \(m\). Apart from the trivial case for \(m=1\), the only known value of \(\lambda_{\mathbb{K}}(m)\) is for \(m=2\) and \(\mathbb{K}=\mathbb{R}\). \textit{B. Grünbaum} [Trans. Am. Math. Soc. 95, 451--465 (1960; Zbl 0095.09002)] conjectured that \(\lambda_{\mathbb{R}}(2)=\frac{4}{3}\) and, \textit{B. L. Chalmers} and \textit{G. Lewicki} [Stud. Math. 200, No. 2, 103--129 (2010; Zbl 1255.46005)] proved it. \textit{G. Basso} [J. Funct. Anal. 277, No. 10, 3560--3585 (2019; Zbl 1443.46004)] delivered the alternative proof of this conjecture. Both proofs are quite complicated, and there was a strong belief that providing an exact value for \(\lambda_{\mathbb{K}}(m)\) in other cases would be a difficult task. In our paper, we present an upper bound of the value \(\lambda_{\mathbb{K}}(m)\), which becomes an exact value for numerous cases. This bound was first stated in [\textit{H. König} and \textit{N. Tomczak-Jaegermann}, J. Funct. Anal. 119, No. 2, 253--280 (1994; Zbl 0818.46015)], but with an erroneous proof. The crucial idea of our proof will be an application of some results from the articles ([\textit{B. Bukh} and \textit{C. Cox}, Isr. J. Math. 238, No. 1, 359--388 (2020; Zbl 1464.94080)] and [\textit{G. Basso}, J. Funct. Anal. 277, No. 10, 3560--3585 (2019; Zbl 1443.46004)]), for which simplified proofs will be given.Optimal recovery from inaccurate data in Hilbert spaces: regularize, but what of the parameter?https://zbmath.org/1528.410972024-03-13T18:33:02.981707Z"Foucart, Simon"https://zbmath.org/authors/?q=ai:foucart.simon"Liao, Chunyang"https://zbmath.org/authors/?q=ai:liao.chunyangSummary: In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied to an explicit model assumption made on the functions to be learned. Working in the framework of Hilbert spaces, this article considers a model assumption based on approximability. It also incorporates observational inaccuracies modeled via additive errors bounded in \(\ell_2\). Earlier works have demonstrated that regularization provides algorithms that are optimal in this situation, but did not fully identify the desired hyperparameter. This article fills the gap in both a local scenario and a global scenario. In the local scenario, which amounts to the determination of Chebyshev centers, the semidefinite recipe of Beck and Eldar (legitimately valid in the complex setting only) is complemented by a more direct approach, with the proviso that the observational functionals have orthonormal representers. In the said approach, the desired parameter is the solution to an equation that can be resolved via standard methods. In the global scenario, where linear algorithms rule, the parameter elusive in the works of Micchelli et al. is found as the byproduct of a semidefinite program. Additionally and quite surprisingly, in case of observational functionals with orthonormal representers, it is established that any regularization parameter is optimal.On property-\((P_1)\) in Banach spaceshttps://zbmath.org/1528.411002024-03-13T18:33:02.981707Z"Thomas, Teena"https://zbmath.org/authors/?q=ai:thomas.teenaSummary: We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by \textit{J. Mach} [J. Approx. Theory 29, 223--230 (1980; Zbl 0467.41015)] as property-\((P_1)\). For a Banach space \(X\), a closed convex subset \(V\) of \(X\) and a subclass \(\mathscr{F}\) of the closed bounded subsets of \(X\), this property, defined for the triplet \((X, V, \mathscr{F})\), describes simultaneous strong proximinality of \(V\) at each of the sets in \(\mathscr{F}\). We establish that if the closed unit ball of a closed subspace of a Banach space \(X\) possesses property-\((P_1)\) for each of the classes of closed bounded, compact and finite subsets of \(X\), then so does the subspace. It is also proved that the closed unit ball of an \(M\)-ideal in an \(L_1\)-predual space satisfies property-\((P_1)\) for the compact subsets of the space. For a Choquet simplex \(K\), we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of \(A(K)\) to satisfy property \((P_1)\) for the compact subsets of \(A(K)\). This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of \(A(K)\) and property-\((P_1)\) of the closed unit ball of the subspace for the compact subsets of \(A(K)\). Further, for a compact Hausdorff space \(S\), a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of \(C(S)\) in terms of property-\((P_1)\) of the subspace and that of its closed unit ball for the compact subsets of \(C(S)\). We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an \(L_1\)-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-\((P_1)\) for the compact subsets of the \(L_1\)-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies \(1 \frac{1}{2} \)-ball property and does not admit restricted Chebyshev center for a closed bounded subset of the Banach space.Some geometrical characterizations of \(L_1\)-predual spaceshttps://zbmath.org/1528.411012024-03-13T18:33:02.981707Z"Thomas, Teena"https://zbmath.org/authors/?q=ai:thomas.teenaSummary: Let \(X\) be a real Banach space. For a non-empty finite subset \(F\) and closed convex subset \(V\) of \(X\), we denote by \(\mathrm{rad}_X(F), \mathrm{rad}_V(F), \mathrm{cent}_X(F)\) and \(d(V, \mathrm{cent}_X(F))\) the Chebyshev radius of \(F\) in \(X\), the restricted Chebyshev radius of \(F\) in \(V\), the set of Chebyshev centers of \(F\) in \(X\) and the distance between the sets \(V\) and \(\mathrm{cent}_X(F)\) respectively. We prove that \(X\) is an \(L_1\)-predual space if and only if for each four-point subset \(F\) of \(X\) and non-empty closed convex subset \(V\) of \(X\),
\[
\mathrm{rad}_V(F) = \mathrm{rad}_X(F) + d(V, \mathrm{cent}_X(F)).
\]
Moreover, we explicitly describe the Chebyshev centers of a compact subset of an \(L_1\)-predual space. Various new characterizations of ideals in an \(L_1\)-predual space are also obtained. In particular, for a compact Hausdorff space \(S\) and a subspace \(\mathcal{A}\) of \(C(S)\) which contains the constant function \(1\) and separates the points of \(S\), we prove that the state space of \(\mathcal{A}\) is a Choquet simplex if and only if \(d(\mathcal{A}, \mathrm{cent}_{C(S)}(F))=0\) for every four-point subset \(F\) of \(\mathcal{A}\). We also derive characterizations for a compact convex subset of a locally convex topological vector space to be a Choquet simplex.Connectedness in asymmetric spaceshttps://zbmath.org/1528.411022024-03-13T18:33:02.981707Z"Tsar'kov, I. G."https://zbmath.org/authors/?q=ai:tsarkov.igor-gSummary: Inverse theorems for geometric approximation theory related to the structure of approximating sets are obtained. Various types of connectedness of sets in asymmetric spaces (generally nonmetrizable) are studied. In particular, conditions are established for an asymmetric space and a subset of this space that guarantee that the intersection of an open ball with this set is path-connected.Uniformly convergent Fourier series with universal power parts on closed subsets of measure zerohttps://zbmath.org/1528.420072024-03-13T18:33:02.981707Z"Khrushchev, Sergey"https://zbmath.org/authors/?q=ai:khrushchev.sergeySummary: Given a closed subset \(E\) of Lebesgue measure zero on the unit circle \(\mathbb{T}\) there is a function \(f\) on \(\mathbb{T}\) with uniformly convergent symmetric Fourier series
\[S_n ( f , \zeta ) = \sum_{k = - n}^n \hat{f} ( k ) \zeta^k {\underset{\mathbb{T}}{\rightrightarrows}} f ( \zeta ),\]
such that for every continuous function \(g\) on \(E\), there is a subsequence of partial power sums
\[S_n^+ ( f , \zeta ) = \sum_{k = 0}^n \hat{f} ( k ) \zeta^k\]
of \(f\), which converges to \(g\) uniformly on \(E\). Here
\[\hat{f} ( k ) = \int_{\mathbb{T}} \overline{\zeta}^k f ( \zeta )\, d m ( \zeta ),\]
and \(m\) is the normalized Lebesgue measure on \(\mathbb{T} \).A new characterization of the Besov spaces associated with Hermite operatorhttps://zbmath.org/1528.420272024-03-13T18:33:02.981707Z"Huang, Jizheng"https://zbmath.org/authors/?q=ai:huang.jizheng"Mo, Huixia"https://zbmath.org/authors/?q=ai:mo.huixiaIn this paper one considers the Hermite operator \(H=-\Delta+|x|^2\) and the associated Hermite-Hölder and Besov spaces. The main result provides a characterization of these spaces by means of the generalized derivatives.
In order to formulate the results one introduces the heat kernel and the Poisson kernel for the operator \(H\). For any \(\gamma>0\), let \(k\) be the smallest integer greater than \(\gamma\). The Hölder space \(\Gamma_H^\gamma\) associated with \(H\) is defined as the set of \(L^\infty\) functions for which there exists a constant \(A_\gamma(f)\) such that
\[
\|d_t^kP_t^Hf\|_\infty\leq A_\infty(f)t^{-k+\gamma},
\]
where \(P_t^H\) is the Poisson kernel associated to \(H\). In order to define high-order Hermite-Hölder spaces, one considers the derivatives associated with \(H\), given by
\[
H = \frac12 \sum_{i=1}^n(A_iA_{-i} + A_{-i}A_i),
\]
where
\[
A_i = d_{x_i} + x_i,\quad A_{-i} = A_i^\ast = -d_{x_i} + x_i,\quad i = 1,2,\dots,n.
\]
Then for each natural number \(k\), the Hermite-Hölder space \(C_H^{k,\alpha}\), \(0<\alpha\leq 1\), consists of the set of functions \(f\) in \(C^k(R^n)\) such that the \(M_\alpha\)-norm of \(f\) plus the sum of the \(M_\alpha\)-norms of \(|A_{i_1}\dots A_{i_m}(f)|\) for \(1\leq |i_1|,\dots,|i_m|\leq n\), \(1\leq m\leq k\) and the sum of the \(\alpha\)-Lipschitz norms of \(|A_{i_1}\dots A_{i_k}(f)|\) for \(1\leq|i_1|,\dots,|i_k|\leq n\), is finite. Here the \(M_\alpha\)-norm of \(f\) means
\[
\|f\|_{M_\alpha}= \sup |\chi(x)^{-\alpha}f(x)|,\ x\in R^n, \quad \text{with }\chi(x) = 1/(1+|x|).
\]
Then one proves the following result:
Let \(f\in L^\infty(R^n)\) and \(0 <\alpha\leq 1\). Then \(f\in C_H^{k,\alpha}(R^n)\) if and only if \(f\in\Gamma_H^\gamma\), where \(\gamma=\alpha+ k\). Moreover the norms of \(f\) in each of these spaces are equivalent.
One gives also a characterization of Besov spaces associated with the operator \(H\) by the Lipschitz condition in a similar way to the classical case.
Reviewer: Julià Cufí (Bellaterra)Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: characterizations of maximal functions, decompositions, and dual spaceshttps://zbmath.org/1528.420282024-03-13T18:33:02.981707Z"Yan, Xianjie"https://zbmath.org/authors/?q=ai:yan.xianjie"He, Ziyi"https://zbmath.org/authors/?q=ai:he.ziyi"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wenThis paper extends the study of the Hardy spaces associated with quasi-Banach function spaces introduced and studied in [\textit{Y. Sawano} et al., Diss. Math. 525, 102 p. (2017; Zbl 1392.42021)] to the Hardy space defined in a space of homogeneous type introduced by \textit{R. R. Coifman} and \textit{G. Weiss} [Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. (Non-commutative harmonic analysis on certain homogeneous spaces. Study of certain singular integrals.). Lect. Notes Math. 242 (1971; Zbl 0224.43006)]. It establishes several real-variable characterizations of Hardy spaces, such as the characterizations in terms of radial and nontangential maximal functions, atomic decompositions, finite atomic decompositions, and molecular characterizations. In addition, it also identifies the dual space of the Hardy space associated with the quasi-Banach function space which is the ball Campanato-type function space.
Reviewer: Kwok Pun Ho (Hong Kong)Realization functionals and description of a modulus of smoothness in variable exponent Lebesgue spaceshttps://zbmath.org/1528.420312024-03-13T18:33:02.981707Z"Volosivets, S. S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: In variable exponent Lebesgue spaces, the equivalence between the modulus of smoothness given by one-sided Steklov means and realization functionals that use Zygmund-Riesz and Euler means is established. A class of functions equivalent to generalized moduli of smoothness of the order \(r \in \mathbb{N}\) is described.Polynomial approximation with respect to multiplicative systems in the Morrey spacehttps://zbmath.org/1528.420322024-03-13T18:33:02.981707Z"Volosivets, S. S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: We establish the main theorems on polynomial approximation with respect to multiplicative systems in the Morrey space and estimate the best approximations and the moduli of smoothness of functions in terms of the growth of the generalized derivatives of their best approximation polynomials, Fourier sums, and Riesz-Zygmund means with respect to multiplicative systems. Also, we provide some criteria for a function to belong to the classes with prescribed majorants for the modulus of smoothness or best approximation in the Morrey space.Stable phase retrieval and perturbations of frameshttps://zbmath.org/1528.420382024-03-13T18:33:02.981707Z"Alharbi, Wedad"https://zbmath.org/authors/?q=ai:alharbi.wedad-r"Freeman, Daniel"https://zbmath.org/authors/?q=ai:freeman.daniel-h-jun"Ghoreishi, Dorsa"https://zbmath.org/authors/?q=ai:ghoreishi.dorsa"Lois, Claire"https://zbmath.org/authors/?q=ai:lois.claire"Sebastian, Shanea"https://zbmath.org/authors/?q=ai:sebastian.shaneaThe authors provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are independent of the dimension of the Hilbert space and the number of vectors in the frame.
Reviewer: Ghanshyam Bhatt (Nashville)Fourier multipliers for Triebel-Lizorkin spaces on compact Lie groupshttps://zbmath.org/1528.430012024-03-13T18:33:02.981707Z"Cardona, Duván"https://zbmath.org/authors/?q=ai:cardona.duvan"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-vSummary: We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.Notes on compactness in \(L^p\)-spaces on locally compact groupshttps://zbmath.org/1528.430022024-03-13T18:33:02.981707Z"Krukowski, Mateusz"https://zbmath.org/authors/?q=ai:krukowski.mateuszAuthor's abstract: ``The main goal of the paper is to provide new insight into compactness in \(L^p\)-spaces on locally compact groups. The article begins with a brief historical overview and the current state of literature regarding the topic. Subsequently, we ``take a step back'' and investigate the Arzelà-Ascoli theorem on a non-compact domain together with one-point compactification. The main idea comes in Section 3, where we introduce the ``\(L^p\)-properties'' (\(L^p\)-boundedness, \(L^p\)-equicontinuity, and \(L^p\)-equivanishing) and study their ``behaviour under convolution''. The paper proceeds with an analysis of Young's convolution inequality, which plays a vital role in the final section. During the ``grand finale'', all the pieces of the puzzle are brought together as we lay down a new approach to compactness in \(L^p\)-spaces on locally compact groups.''
Various locally compact groups in the context of the main result are examined. The author's approach is new on the issues of compactness in \(L^p\)-spaces on locally compact groups.
Reviewer: Serap Öztop (İstanbul)Tikhonov regularisation for the Weierstrass transform associated with the Kontorovich-Lebedev transformhttps://zbmath.org/1528.440012024-03-13T18:33:02.981707Z"Dziri, Moncef"https://zbmath.org/authors/?q=ai:dziri.moncef"Kroumi, Anis"https://zbmath.org/authors/?q=ai:kroumi.anisSummary: Using the best approximations and the theory of reproducing kernels, we will give real inversion formulas for the Weierstrass transform associated with the Kontorovich-Lebedev transform, and we will check the estimates of extremal functions.Function spaces and dense approximation. Proceedings of the conference, University of Bonn, Bonn, Germany, September 30 -- October 3, 1974https://zbmath.org/1528.460012024-03-13T18:33:02.981707ZThe articles of this volume will be reviewed individually within the series [Bonn. Math. Schr. 81 (1975)].
Indexed articles:
\textit{Schmets, Jean}, Simple and weak compactnesses in spaces of continuous functions [Zbl 0328.46019]
\textit{Schmets, Jean; Zafarani, Jafar}, A weak strict topology and discrete measures [Zbl 0328.46020]
\textit{Prolla, Joao B.}, Dense approximation for polynomial algebras [Zbl 0328.46033]
\textit{Aron, Richard M.}, The approximation property for spaces of holomorphic functions on a Banach space [Zbl 0332.46032]
\textit{Bierstedt, Klaus Dieter}, The approximation property for weighted function spaces [Zbl 0333.46023]
\textit{Bierstedt, Klaus Dieter}, Tensor products of weighted spaces [Zbl 0333.46024]
\textit{Bierstedt, Klaus Dieter; Gramsch, Bernhard; Meise, Reinhold}, Lokalkonvexe Garben und gewichtete induktive Limites \(\mathcal F\)-morpher Funktionen [Zbl 0335.46018]
\textit{Haydon, R. G.}, On the Stone Weierstrass theorem for the strict and superstrict topologies [Zbl 0337.46024]
\textit{Hogbe-Nlend, H.}, Topologies et bornologies d'approximation linéaire [Zbl 0338.46003]
\textit{Kleinstück, Gert}, Duals of weighted spaces of continuous functions [Zbl 0345.46023]
\textit{de la Fuente, Angel}, The Weierstrass-Stone theorem for Clifford and Cayley-Dickson algebras [Zbl 0363.41033]Kernel theorems for Beurling-Björck type spaceshttps://zbmath.org/1528.460022024-03-13T18:33:02.981707Z"Neyt, Lenny"https://zbmath.org/authors/?q=ai:neyt.lenny"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonThe authors consider a general class of Beurling-Björck spaces (of Beurling and Roumieu type) defined in terms of two weight function systems \({\mathcal V}\) and \({\mathcal W}\). In the case that \({\mathcal V} = \{e^{\frac{1}{\lambda}v}: \lambda > 0\}\) and \({\mathcal W} = \{e^{\frac{1}{\lambda}w}: \lambda > 0\}\) for appropriate non-negative continuous functions \(v, w\) on \({\mathbb R}^d\), the classical Beurling-Björck spaces are recovered. Section~3 (along with Appendix~A) characterizes when the general Beurling-Björck spaces are nuclear in terms of the weight systems, while Section~4 is devoted to the new kernel theorems. An important tool in the proof of the kernel theorem for Roumieu-type spaces is the projective description of said spaces using maximal Nachbin families associated with weight systems.
Reviewer: Antonio Galbis (València)Domain of generalized Riesz difference operator of fractional order in Maddox's space \(\ell(p)\) and certain geometric propertieshttps://zbmath.org/1528.460032024-03-13T18:33:02.981707Z"Yaying, Taja"https://zbmath.org/authors/?q=ai:yaying.taja"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Mohiuddine, S. A."https://zbmath.org/authors/?q=ai:mohiuddine.syed-abdulFrom the preface: This chapter introduces the paranormed Riesz difference sequence space of fractional order obtained by the domain of generalized difference operator in Maddox's space $\ell(p)$, and obtains some topological and geometric properties, and Schauder basis of this sequence space. The $\alpha$-, $\beta$- and $\gamma$-duals and characterization of some matrix classes on the new space are presented.
For the entire collection see [Zbl 1495.40002].Transfinite almost square Banach spaceshttps://zbmath.org/1528.460042024-03-13T18:33:02.981707Z"Avilés, Antonio"https://zbmath.org/authors/?q=ai:aviles.antonio"Ciaci, Stefano"https://zbmath.org/authors/?q=ai:ciaci.stefano"Langemets, Johann"https://zbmath.org/authors/?q=ai:langemets.johann"Lissitsin, Aleksei"https://zbmath.org/authors/?q=ai:lissitsin.aleksei"Rueda Zoca, Abraham"https://zbmath.org/authors/?q=ai:rueda-zoca.abrahamLet \(X\) be a Banach space with unit ball \(B_X\) and unit sphere \(S_X\). If, for every finite dimensional subspace \(Y\) of \(X\) and \(\varepsilon > 0\), there exists \(y \in S_X\) such that \(\|y + r x\| \le (1+\varepsilon)\max(\|y\|,|r|)\) for all \(y \in Y\) and \(r \in \mathbb{R}\), we say that \(X\) is almost square.
It is known that a Banach space contains an isomorphic copy of \(c_0\) if and only if it can be equivalently renormed to be almost square. The authors introduce and study transfinite versions of almost square Banach spaces with the aim of relating them to the containment of isomorphic copies of \(c_0(\kappa)\), where \(\kappa\) is some uncountable cardinal.
Given a set \(A\) and a cardinal \(\kappa\), denote by \(\mathcal{P}_\kappa(A)\) and \(\mathcal{P}_{< \kappa}(A)\) the sets of all subsets of \(A\) of cardinality at most \(\kappa\) and strictly less than \(\kappa\), respectively.
A Banach space \(X\) is \({{<}\kappa}\)-almost square (ASQ\(_{<\kappa}\)) if, for every set \(A \in \mathcal{P}_{< \kappa}(S_X)\) and \(\varepsilon > 0\), there exists \(y \in S_X\) such that \(\|x \pm y\| \le 1+ \varepsilon\) for all \(x \in A\). If we can drop the \(\varepsilon\), we say that \(X\) is \({<}\kappa\)-square (SQ\(_{< \kappa}\)).
The authors provide several examples and stability results for these properties by taking direct sums, tensor products and ultraproducts. Let us mention a few results.
It is shown that, if \(K\) is a \(T_4\) locally compact space and \(\kappa\) is an uncountable cardinal, then \(C_0(K)\) is SQ\(_{< \kappa}\) if and only if it is ASQ\(_{< \kappa}\). An example of a Banach space \(X\) is given for every infinite cardinal \(\kappa\) such that \(X\) is ASQ\(_{< \kappa}\), but \(X^*\) does not contain \(\ell_1(\omega_1)\) and, in particular, \(X\) does not contain \(c_0(\omega_1)\). But, if \(K\) is a compact Hausdorff space and \(C(K)\) admits an equivalent ASQ\(_{< \kappa}\) norm, then \(C(K)\) contains an isomorphic copy of \(c_0(\kappa)\). Finally, we mention the result that, if \(\kappa\) is an infinite cardinal of uncountable cofinality and \(X\) is a Banach space of density character \(\kappa\) containing an isomorphic copy of \(c_0(\kappa)\), then \(X\) admits an equivalent SQ\(_{< \kappa}\) renorming.
Transfinite analogues of the (symmetric) strong diameter two property and octahedrality are also studied.
Reviewer: Vegard Lima (Kristiansand)Dynamics of the semigroup of contractive automorphisms of Banach spaceshttps://zbmath.org/1528.460052024-03-13T18:33:02.981707Z"Cabello Sánchez, Félix"https://zbmath.org/authors/?q=ai:cabello-sanchez.felix"Cabello Sánchez, Javier"https://zbmath.org/authors/?q=ai:cabello-sanchez.javierTheorem 2.3, Proposition 3.1 and Theorem 4.2 in the paper rather clearly summarize the contents of this interesting slightly out-of-the-box paper devoted, as the title says, to explore the dynamics of the semigroup of contractive automorphisms \(\mathrm{Aut}_1(X)\) when acting on the unit sphere \(S\) of a real Banach space \(X\). What the title does not say is that this \(X\) will be, for most of the paper, finite-dimensional. Three notions have to be considered: \textit{semitransitivity} (for each \(x,y\in S\) there is a contractive automorphism \(T\) such that \(Tx=y\)), \textit{bounded semitransitivity} (there is a constant \(K\) such that all \(T\) appearing in the transitivity notion satisfy \(\|T^{-1}\|\leq K\)) and \textit{uniform micro semitransitivity} (for every \(\varepsilon>0\) there is \(\delta>0\) such that, whenever \(x,y\in S\) with \(\|x-y\|<\delta\), there is a contractive automorphism \(T\) such that \(Tx=y\) and \(\|T - \mathrm{Id}\|<\varepsilon\)). These three pieces, ST, BST and UMST in short, articulate the dynamics of \(\mathrm{Aut}_1(X)\) on a finite-dimensional \(X\) as follows.
\begin{itemize}
\item \(X\) is ST (resp. BST) if and only if \(X^*\) is ST (resp. BST).
\item \(X\) is ST (resp. BST) if and only if every point in the unit sphere admits an inner and outer ellipsoid (resp. there is \(\lambda\) such that every point in the unit sphere admits an inner ellipsoid \(E\) and an outer ellipsoid \(F\) such that \(F\subset \lambda E\)).
\item If \(\dim X=2\), then \(X\) is UMST if and only if its unit sphere is of class \(C^2\) and has strictly positive curvature at every point.
\end{itemize}
These results do not exhaust the paper: the reader will find many additional clever examples and a description of UMST norms on the plane.
Reviewer: Jesús M. F. Castillo (Badajoz)On the Crawford number attaining operatorshttps://zbmath.org/1528.460062024-03-13T18:33:02.981707Z"Choi, Geunsu"https://zbmath.org/authors/?q=ai:choi.geunsu"Lee, Han Ju"https://zbmath.org/authors/?q=ai:lee.han-ju|lee.han-ju.1The Crawford number \(c(T)\) of the bounded linear operator \(T\) defined on \(X\) is defined to be the following infimum
\[
c(T) := \inf \Big\{ |x^*(T(x))|: (x, x^*) \in \Pi(X) \Big\}
\]
where \(\Pi(X):= \{ (x, x^*) \in S_{X} \times S_{X^*}: x^*(x) = 1\}\). In the present paper, the authors study the subset \(\mbox{CNA}(X)\) of all bounded linear operators which attain their Crawford number. Among other results, they prove that every operator attains its Crawford number when \(X\) is finite-dimensional; they show that \(c(T) = c(T^*)\) for every operator \(T \in \mathcal{L}(X)\) and give a characterization of when \(T^*\) belongs to \(\mbox{CNA}(X^*)\).
They also study the density of Crawford number attaining operators. Indeed, after proving that there exists an injective operator \(T\) with \(c(T) = 0\) which never attains its Crawford number, they start studying conditions on the involved Banach spaces \(X\) to get such a denseness. For instance, whenever \(c(T) = 0\), the operator \(T\) can be approximated by Crawford number attaining operators; for every Banach space \(X\), the subset of all Crawford number attaining compact operators is always dense in \(\mathcal{K}(X)\); moreover, when \(X\) has the RNP, the subset \(\mbox{CNA}(X)\) is always dense in \(\mathcal{L}(X)\) for every Banach space \(X\).
They conclude the paper with Bollobás type theorems for Crawford numbers and several interesting questions about the topic.
Reviewer: Sheldon Dantas (València)On the transfinite symmetric strong diameter two propertyhttps://zbmath.org/1528.460072024-03-13T18:33:02.981707Z"Ciaci, Stefano"https://zbmath.org/authors/?q=ai:ciaci.stefanoA Banach space \(X\) has the symmetric strong diameter two property (SSD2P) if, for every \(x_1^*,x_2^*,\ldots,x_n^* \in S_{X^*}\) and \(\varepsilon > 0\), there are \(x_1,\ldots,x_n,y \in B_X\) such that \(\|y\| \ge 1 - \varepsilon\), \(x_i \pm y \in B_X\) and \(x_i(x_i) \ge 1 - \varepsilon\) for all \(1 \le i \le n\).
In this paper, the author studies generalizations of the SSD2P where, instead of a set of finite cardinality we start with a larger set. These generalizations with slightly different notation first appeared in [\textit{A. Avilés} et al., Stud. Math. 271, No. 1, 39--63 (2023; Zbl 1528.46004)].
Let \(r \in (0,1)\), \(B \subset B_X\) and \(A \subset S_{X^*}\). We say that \(B\) \(r\)-norms \(A\) if, for every \(x^* \in A\), there is an \(x \in B\) such that \(x^*(x) \ge r\). If this holds for all \(r \in (0,1)\), \(B\) norms \(A\).
For a Banach space \(X\) and an infinite cardinal \(\kappa\), we say that \(X\) has the SSD2P\(_\kappa\) if, for every set \(A\subset S_{X^*}\) of cardinality \(< \kappa\) and \(\varepsilon > 0\), there are \(B \subset B_X\) which \((1-\varepsilon)\)-norms \(A\), and \(y \in B_X\) with \(\|y\| \ge 1 - \varepsilon\) satisfying \(B \pm y \in B_X\).
A \(1\)-norming and attaining version of the SSD2P\(_\kappa\) is also studied: \(X\) has the \(1\)-ASSD2P\(_\kappa\) if, for every set \(A \subset S_{X^*}\) of cardinality \(< \kappa\), there are \(B \subset S_X\) which norms \(A\), and \(y \in S_X\) satisfying \(B\pm y \subset S_X\).
Let \(X\) and \(Y\) be Banach spaces and \(\kappa > \aleph_0\). It is shown that \(X\) or \(Y\) has the SSD2P\(_\kappa\) if and only if \(X \oplus_\infty Y\) has the SSD2P\(_\kappa\). General \(\infty\)-sums are also studied. It is also shown that, if \(X\) and \(Y\) have the SSD2P\(_\kappa\), then the projective tensor product \(X \hat{\otimes}_\pi Y\) has the SSD2P\(_\kappa\). However, there are characterizations of the SSD2P which cannot be transferred to the transfinite setting and the proof of the latter result is different from the proof for the SSD2P. Positive results are given for \(C_0(X)\) spaces where \(X\) is a locally compact Hausdorff space, but even in this case the situation is not entirely clear.
Reviewer: Vegard Lima (Kristiansand)On the quasi-Figiel problem and extension of \(\varepsilon\)-isometry on unit sphere of \(\mathcal{L}_{\infty, 1^+}\) spacehttps://zbmath.org/1528.460082024-03-13T18:33:02.981707Z"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.rui|liu.rui.1"Yin, Jifu"https://zbmath.org/authors/?q=ai:yin.jifuLet \(X\) be a real Banach space. \(X\) is called an \(\mathcal{L}_{\infty,1^+}\) space if the following holds: for every finite-dimensional subspace \(U\subseteq X\) and every \(\varepsilon>0\), there exists a finite-dimensional subspace \(V\subseteq X\) such that \(U\subseteq V\) and \(d(V,\ell_{\infty}^n)\leq 1+\varepsilon\), where \(d\) is the Banach-Mazur distance and \(n=\dim(V)\).
The main result of the paper is the following: if \(X\) is an \(\mathcal{L}_{\infty,1^+}\) space, \(Y\) is another Banach space, and \(T:S_X \rightarrow S_Y\) is a bijective \(\varepsilon\)-isometry between the unit spheres \(S_X\) and \(S_Y\), then there is a bijective \(5\varepsilon\)-isometry \(\tilde{T}:B_X \rightarrow B_Y\) between the unit balls \(B_X\) and \(B_Y\) such that \(\tilde{T}|_{S_X}=T\).
Reviewer: Jan-David Hardtke (Leipzig)Random \(\delta\)-nearsurjective \(\varepsilon\)-isometries on random normed moduleshttps://zbmath.org/1528.460092024-03-13T18:33:02.981707Z"Wang, Yachao"https://zbmath.org/authors/?q=ai:wang.yachao"Guo, Tiexin"https://zbmath.org/authors/?q=ai:guo.tiexinSummary: Let \((\Omega,\mathcal{F},P)\) be a probability space, \(\mathbb R\) the scalar field of real numbers, \(L^0(\mathcal{F},\mathbb R)\) the equivalence classes of \(\mathbb R\)-valued \(\mathcal{F}\)-measurable random variables on \(\Omega\), \((E,\|\cdot\|)\) and \((F,\|\cdot\|)\) two complete random normed modules over \(\mathbb R\) with base \((\Omega,\mathcal{F},P)\). The main theorem of this paper is the following approximation result for random \(\delta\)-nearsurjective \(\varepsilon\)-isometries between random normed modules: if \(f:E\to F\) is a stable random \(\delta\)-nearsurjective \(\varepsilon\)-isometry with \(f(0)=0\), where \(\varepsilon,\delta\in L^0(\mathcal{F},\mathbb R)\) and \(\varepsilon,\delta\ge0\), then there exists a surjective \(L^0\)-linear random isometry \(U\) between \(E\) and \(F\) such that \(\|f(x)-U(x)\|\le 4\varepsilon\) for all \(x\in E\). Furthermore, making use of the above result and the relations between random normed modules and classical normed spaces, we give the approximation result for sample-continuous random operators: let \((X,\|\cdot\|)\) and \((Y,\|\cdot\|)\) be two real separable Banach spaces, \(\varepsilon^0\) and \(\delta^0\) two nonnegative random variables and \(f:\Omega\times X\to Y\) a random operator such that \(f(\omega,\cdot):X\to Y\) is a continuous \(\delta^0(\omega)\)-nearsurjective \(\varepsilon^0(\omega)\)-isometry and \(f(\omega,0)=0\) for any \(\omega\in\Omega\), then there exist a sample-linear and almost everywhere (briefly, a.e.) isometric random operator \(U:\Omega\times X\to Y\) and \(\Omega_0\in\mathcal{F}\) with \(P(\Omega_0)=1\) such that \(\|f(\omega,x)-U(\omega,x)\|\le 4\varepsilon^0(\omega)\), \(\forall(\omega,x)\in\Omega_0\times X\). It is the first time that sample-linear and a.e. isometric random operators are used to approximate sample-continuous nonlinear random operators.Projections and unconditional bases in direct sums of \(\ell_p\) spaces, \(0 < p \leq \infty\)https://zbmath.org/1528.460102024-03-13T18:33:02.981707Z"Albiac, Fernando"https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José Luis"https://zbmath.org/authors/?q=ai:ansorena.jose-luisElaborating on earlier work of \textit{I.~S. Edelstein} and \textit{P.~Wojtaszczyk} [Stud. Math. 56, 263--276 (1976; Zbl 0362.46017)] and several contributions by N.~J. Kalton about complemented subspaces and uniqueness of unconditional bases, the authors solve a question posed by \textit{A.~Ortynski} [Math. Nachr. 103, 109--116 (1981; Zbl 0492.46006)]. Namely, it is shown that every basis of a finite sum of \(\ell_p\) spaces for different \(p\in(0,\infty)\), possibly in combination with \(c_0\), splits into unconditional bases of each summand. The authors deal with the apparently harder case of \(\ell_1\) appearing in the finite sum, which is handled here using the notion of anti-euclidean quasi-Banach lattice from the work of \textit{P.~G. Casazza} and \textit{N.~J. Kalton} [Isr. J. Math. 103, 141--175 (1998; Zbl 0939.46009)]. The paper is very well written, as is customary with the authors.
Reviewer: Pedro Tradacete (Madrid)Generalized Sierpinski functions and fragmentable compact spaces.https://zbmath.org/1528.460112024-03-13T18:33:02.981707Z"Matsuda, Minoru"https://zbmath.org/authors/?q=ai:matsuda.minoru(no abstract)Estimates of the Chebyshev radius in terms of the MAX-metric function and the MAX-projection operatorhttps://zbmath.org/1528.460122024-03-13T18:33:02.981707Z"Tsar'kov, I. G."https://zbmath.org/authors/?q=ai:tsarkov.igor-gSummary: Singleton approximations of sets in asymmetric spaces are studied. Estimate of the Chebyshev radius of a set depending on the behavior of the MAX-distance function and if the MAX-projection operator has not too many points of discontinuity.On the property (C) of Corson and other sequential properties of Banach spaceshttps://zbmath.org/1528.460132024-03-13T18:33:02.981707Z"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Poveda, Alejandro"https://zbmath.org/authors/?q=ai:poveda.alejandroThis paper deals with several properties of Banach spaces whose definition involves the analysis of their convex subsets. A Banach space is said to have \textit{Corson's property (C)} if for every family of closed convex subsets with empty intersection, there exists a countable subfamily that already has empty intersection. The authors consider (at least) two more properties, analogous to the topological characterization of closed sets and points in the closure by means of convergent sequences, but involving convex subsets. A space has \textit{property \(\mathcal E\)} if for convex subsets one can characterize belonging to the weak\(^*\)-closure by means of being the weak\(^*\)-limit of a sequence in the set; on the other hand, a space has \textit{property \(\mathcal E'\)} if weak\(^*\)-sequentially closed convex subsets of the dual ball are weak\(^*\)-closed (thus it is not hard to see that property \(\mathcal E'\) is a weakening of property \(\mathcal E\)).
After the introduction (the first section), the second section of the present paper contains the proof that property \(\mathcal E'\) implies Corson's property (C); this section also features a proof of a result (Theorem~3 in the paper), stemming from the PhD thesis of the first author, that property \(\mathcal E'\) implies that the dual ball is weak\(^*\)-block compact. Finally, the third section contains one of the core results of the paper, namely that under \(\mathsf{PFA}\) Corson's property (C) implies that the unit ball of the dual, equipped with the weak\(^*\) topology, has countable tightness. It remains open whether such an implication holds in \(\mathsf{ZFC}\) alone.
At the end of the paper, putting all of the results therein together with a result of \textit{Z.~Balogh} [Proc. Am. Math. Soc. 105, No.~3, 755--764 (1989; Zbl 0687.54006)], the conclusion is that four conditions are equivalent under \(\mathsf{PFA}\): Corson's property (C), property \(\mathcal E\), the unit ball of the dual having countable tightness under the weak\(^*\) topology, and having a weak\(^*\) sequential dual ball. It is worth noting that this combines with a result of \textit{C.~Brech} [Construções genéricas de espaços de Asplund \(C(K)\). Universidade de São Paulo and Université Paris VII (PhD Thesis) (2008)], claiming the consistency that property (C) does not imply property \(\mathcal E\) -- but establishing, in fact, that property (C) does not imply property \(\mathcal E'\). Therefore one may conclude that the statement that property (C) implies property \(\mathcal E'\) is independent of \(\mathsf{ZFC}\).
Reviewer: David J. Fernández-Bretón (Ciudad de México)Operators with the Lipschitz bounded approximation propertyhttps://zbmath.org/1528.460142024-03-13T18:33:02.981707Z"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.rui"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie|shen.jie.2|shen.jie.3|shen.jie.4|shen.jie.1|shen.jie.5"Zheng, Bentuo"https://zbmath.org/authors/?q=ai:zheng.bentuoThe present paper is dedicated to the Lipschitz bounded approximation property for bounded linear operators. The authors show that whenever an operator can be approximated by a net of uniformly bounded finite rank Lipschitz mappings pointwise, then it can also be approximated by a net of uniformly bounded finite rank linear operators under the strong operator topology. As a consequence of it, they recover a theorem due to \textit{G.~Godefroy} and \textit{N.~J. Kalton} [Stud. Math. 159, No.~1, 121--141 (2003; Zbl 1059.46058)]
which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space. Another application of their main result implies that a Banach space has an (unconditional) Lipschitz frame if and only if it has an (unconditional) Schauder frame.
Reviewer: Sheldon Dantas (València)On separability of the unbounded norm topologyhttps://zbmath.org/1528.460152024-03-13T18:33:02.981707Z"Kandić, M."https://zbmath.org/authors/?q=ai:kandic.marko"Vavpetič, A."https://zbmath.org/authors/?q=ai:vavpetic.alesSummary: In this paper we continue the investigation of topological properties of the unbounded norm ($un$-)topology in normed lattices. We characterize separability and second countability of the $un$-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space \(X\) over a semi-finite measure space is separable if and only if it has a \(\sigma\)-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the $un$-topology.Regular measures of noncompactness and Ascoli-Arzelà type compactness criteria in spaces of vector-valued functionshttps://zbmath.org/1528.460162024-03-13T18:33:02.981707Z"Caponetti, Diana"https://zbmath.org/authors/?q=ai:caponetti.diana"Trombetta, Alessandro"https://zbmath.org/authors/?q=ai:trombetta.alessandro"Trombetta, Giulio"https://zbmath.org/authors/?q=ai:trombetta.giulioThe authors study measures of noncompactness in spaces of vector-valued bounded functions and vector-valued bounded differentiable functions. They also introduce a new version of equicontinuity and obtain compactness criteria similar to the classical Ascoli-Arzelà theorem.
Reviewer: Jan-David Hardtke (Leipzig)Hamming graphs and concentration properties in non-quasi-reflexive Banach spaceshttps://zbmath.org/1528.460172024-03-13T18:33:02.981707Z"Fovelle, A."https://zbmath.org/authors/?q=ai:fovelle.audrey|fovelle.aThe author studies concentration properties of Lipschitz maps from a Hamming graph into a Banach space with emphasis on their stability under sums of Banach spaces.
Besides studying the known properties \(\lambda\)-\(HFC_p\) (Hamming full concentration) and \(\lambda\)-\(HIC_p\) (Hamming interlaced concentration), the author introduces their directional variants \(\lambda\)-\(HFC_{p,d}\) and \(\lambda\)-\(HIC_{p,d}\).
The main results are the following:
{Theorem:} Let \((X_n)\) be a sequence of Banach spaces each having the property \(\lambda\)-\(HIC_{p,d}\) and let \(E\) be a reflexive Banach space with 1-unconditional \(p\)-convex basis with convexity constant 1. Then for every \(\epsilon > 0\) the sum \(X = (\sum X_n)_E\) has the property \((\lambda + 2 + \epsilon)\)-\(HIC_{p,d}\).
A consequence of this theorem is that an \(\ell_q\) sum of quasi-reflexive Banach spaces satisfying upper \(\ell_p\) tree estimates cannot equi-Lipschitz contain the Hamming graphs, which provides the first non-quasi-reflexive example of such space.
{Theorem:} Every Banach space with the property \(HIC_{\infty}\) is asymptotically \(c_0\).
As a consequence, the Banach space \(T^*(T^*)\) (where \(T^*\) is Tsirelson's original space and \(T^*(T^*)\) stands for the \(T^*\)-sum of \(T^*\)) does not have the property \(HIC_{\infty}\) while having all the properties \(HFC_{p,d}\) for \(1<p<\infty\).
It is known that every quasi-reflexive asymptotically \(c_0\) Banach space has property \(HIC_{\infty}\), so this result gives us a partial converse. It is not known if the property \(HIC_{\infty}\) implies quasi-reflexivity.
The rest of the paper is devoted to a construction, due to Schlumprecht, of an asymptotically \(c_0\) dual Banach space without any of the concentration properties mentioned.
Reviewer: Zdeněk Silber (Warszawa)Compact Hölder retractions and nearest point mapshttps://zbmath.org/1528.460182024-03-13T18:33:02.981707Z"Medina, Rubén"https://zbmath.org/authors/?q=ai:medina.rubenSummary: In this paper, an \(\alpha\)-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive \(\alpha < 1\). This constitutes a positive solution to a Hölder version of a question raised by \textit{G.~Godefroy} and \textit{N.~Ozawa} [Proc. Am. Math. Soc. 142, No.~5, 1681--1687 (2014; Zbl 1291.46013)].
In fact, compact convex sets are found to be absolute \(\alpha\)-Hölder retracts under certain assumption of flatness.Spaces and non-exact groups admitting a coarse embedding in a Hilbert space [after Arzhantseva, Guentner, Osajda, Špakula]https://zbmath.org/1528.460192024-03-13T18:33:02.981707Z"Khukhro, Ana"https://zbmath.org/authors/?q=ai:khukhro.anaSummary: In the study of metric spaces, it is often the coarse geometric structure that plays a crucial role. Geometric group theory has helped us effectively study groups as geometric objects via their Cayley graphs and since, coarse geometric properties of groups have had profound implications for several important conjectures in topology and analysis. One way to create examples of groups of interest for these conjectures is to use small cancellation theory to embed sequences of finite graphs whose coarse geometry we can control into the Cayley graphs of groups. To achieve this, it is vital to have a rich source of examples of sequences of finite graphs with certain properties. These can also be constructed with the help of groups, by taking a sequence of Cayley graphs of finite quotients of a group, and exploiting the connections between group-theoretic properties and geometric properties of these graphs. Such a construction of Arzhantseva-Guentner-Špakula, involving covering spaces and spaces with walls, has been used by Osajda, together with previous work of Arzhantseva-Osajda, to prove the existence of non-exact groups admitting a coarse embedding into a Hilbert space.
For the entire collection see [Zbl 1456.00108].The set of \(p\)-adic continuous functions not satisfying the Luzin (N) propertyhttps://zbmath.org/1528.460202024-03-13T18:33:02.981707Z"Fernández-Sánchez, J."https://zbmath.org/authors/?q=ai:fernandez-sanchez.juan"Maghsoudi, S."https://zbmath.org/authors/?q=ai:maghsoudi.saeid"Rodríguez-Vidanes, D. L."https://zbmath.org/authors/?q=ai:rodriguez-vidanes.daniel-l"Seoane-Sepúlveda, J. B."https://zbmath.org/authors/?q=ai:seoane-sepulveda.juan-benignoSummary: In this short note, we will study the existence of a vector space of continuous functions \(f:\mathbb{Z}_p \rightarrow \mathbb{Q}_p\), where \(\mathbb{Z}_p\) and \(\mathbb{Q}_p\) are, respectively, the ring of \(p\)-adic integers and the field of \(p\)-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.Universality theorems for asymmetric spaceshttps://zbmath.org/1528.460212024-03-13T18:33:02.981707Z"Alimov, A. R."https://zbmath.org/authors/?q=ai:alimov.alexey-rSummary: Spaces with asymmetric metric and asymmetric norm are considered. It is shown that any metrizable separable asymmetrically normed linear space \((X,\Vert\cdot|)\) can be isometrically isomorphically imbedded, as an affine linear manifold, into the classical space \(C[0,1]\) with uniform norm \(\Vert\cdot \Vert_C\). A similar result is obtained for spaces of density \(\mathfrak{a}\). For spaces with asymmetric metric, it is shown that each such space of density \(\mathfrak{a}\) is isometric to a part of the space \(C([0,1]^{\mathfrak{a}})\) with the asymmetric seminorm \(p(f)=\Vert f_+ \Vert_C\), where \(f_+ (t)=\max\{f(t),0\}\).Power dilation systems \(\{f(z^k)\}_{k\in \mathbb{N}}\) in Dirichlet-type spaceshttps://zbmath.org/1528.460222024-03-13T18:33:02.981707Z"Dan, H."https://zbmath.org/authors/?q=ai:dan.haitao|dan.hiroshige|dan.hui|dan.hiroshi"Guo, K."https://zbmath.org/authors/?q=ai:guo.kejian|guo.kongming|guo.kaiwen|guo.kanghui|guo.kaihang|guo.keyu|guo.kaiyang|guo.kai|guo.kailing|guo.kaili|guo.konghui|guo.kevin|guo.kang|guo.kunyi|guo.kairui|guo.katherine|guo.kehua|guo.kangxian|guo.kui|guo.kunqi|guo.kailin|guo.kuanliang|guo.karen|guo.kun|guo.kangquan|guo.kunyu|guo.keke|guo.konghua|guo.kan|guo.kaizhong|guo.krystal|guo.ke|guo.kexin|guo.keli|guo.kaiyuan|guo.kaihong|guo.kejun|guo.kailun|guo.keyiSummary: Power dilation systems \(\{f(z^k)\}_{k\in \mathbb{N}}\) in Dirichlet-type spaces \(\mathcal{D}_t\) \( (t\in \mathbb{R})\) are treated. When \(t\neq 0\), it is proved that a system of functions \(\{f(z^k)\}_{k\in \mathbb{N}}\) is orthogonal in \(\mathcal{D}_t\) only if \(f=cz^N\) for some constant \(c\) and some positive integer \(N\). Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.Totally null sets and capacity in Dirichlet type spaceshttps://zbmath.org/1528.460232024-03-13T18:33:02.981707Z"Chalmoukis, Nikolaos"https://zbmath.org/authors/?q=ai:chalmoukis.nikolaos"Hartz, Michael"https://zbmath.org/authors/?q=ai:hartz.michaelAuthors' abstract: In the context of Dirichlet type spaces on the unit ball of \(\mathbb{C}^d\), also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthening of boundary interpolation theorems of \textit{V.~V. Peller} and \textit{S.~V. Khrushchev} [Usp. Mat. Nauk 37, No.~1(223), 53--124 (1982; Zbl 0497.60033)] and of
\textit{W.~S. Cohn} and \textit{I.~E. Verbitsky} [Mich. Math. J. 42, No.~1, 79--97 (1995; Zbl 0833.31007)].
Reviewer: Şerban Costea (Piteşti)Higher-order de Branges-Rovnyak and sub-Bergman spaceshttps://zbmath.org/1528.460242024-03-13T18:33:02.981707Z"Gu, Caixing"https://zbmath.org/authors/?q=ai:gu.caixing"Hwang, In Sung"https://zbmath.org/authors/?q=ai:hwang.in-sung"Lee, Woo Young"https://zbmath.org/authors/?q=ai:lee.woo-young"Park, Jaehui"https://zbmath.org/authors/?q=ai:park.jaehuiSummary: The sub-Bergman spaces are de Branges-Rovnyak subspaces of Bergman space \(A^2\) defined by the contraction \(T_b\) or \(T_b^\ast\) for an analytic symbol \(b\). The fact that both \(T_b\) and \(T_b^\ast\) are 2-hypercontractions on \(A^2\) leads to the introduction of a new type of sub-Bergman spaces, which will be called higher-order sub-Bergman spaces. We show these new spaces are different and yet connected in a nice way with the sub-Bergman spaces. The close relationship of these new spaces to the original de Branges-Rovnyak subspaces of the Hardy spaces is also explored. A similar study is conducted on weighted Bergman spaces \(A_\alpha^2\) where both \(T_b\) and \(T_b^\ast\) are \([\alpha + 2]\)-hypercontractions.Multipliers for global Morrey spaceshttps://zbmath.org/1528.460252024-03-13T18:33:02.981707Z"Berezhnoi, Evgenii I."https://zbmath.org/authors/?q=ai:berezhnoi.evgenii-iSummary: Based on a new approach for a wide class of global Morrey spaces, we give an exact description of the multiplier space between two Morrey spaces from this class. It is shown that in this case the multiplier space for a couple of Morrey spaces is an approximation Morrey space structurally constructed from the original spaces.Commutators of classical operators in a new vanishing Orlicz-Morrey spacehttps://zbmath.org/1528.460262024-03-13T18:33:02.981707Z"Deringoz, Fatih"https://zbmath.org/authors/?q=ai:deringoz.fatih"Dorak, Kendal"https://zbmath.org/authors/?q=ai:dorak.kendal"Mislar, Farah"https://zbmath.org/authors/?q=ai:mislar.farah-alissaSummary: We study mapping properties of commutators of classical operators of harmonic analysis -- commutators of maximal, singular and fractional operators in a new vanishing subspace of Orlicz-Morrey spaces. We show that the vanishing property defining that subspace is preserved under the action of those operators.Multidimensional analogs of theorems about the density of sums of shifts of a single functionhttps://zbmath.org/1528.460272024-03-13T18:33:02.981707Z"Dyuzhina, N. A."https://zbmath.org/authors/?q=ai:dyuzhina.n-aThe paper discusses multidimensional versions of the results in the articles [\textit{P.~A. Borodin}, Izv. Math. 81, No.~6, 1080--1094 (2017; Zbl 1384.41016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 81, No.~6, 23--37 (2017)], [\textit{P.~A. Borodin} and \textit{S.~V. Konyagin}, Anal. Math. 44, No.~2, 163--183 (2018; Zbl 1413.42002)].On basicity of perturbed system of exponents in grand-Lebesgue spaceshttps://zbmath.org/1528.460282024-03-13T18:33:02.981707Z"Ismailov, Migdad I."https://zbmath.org/authors/?q=ai:ismailov.migdad-i"Alili, Vafa Q."https://zbmath.org/authors/?q=ai:alili.vafa-q"Aliyarova, Ilakha F."https://zbmath.org/authors/?q=ai:aliyarova.ilakha-fSummary: This work is dedicated to the study of basicity of perturbed system of exponents \(\left\{e^{i(n - \beta \operatorname{sign}n)t}\right\}_{n\in \mathbb Z}\) in grand-Lebesgue spaces \(L_{p)}(-\pi, \pi)\), where \(\beta\) is a complex parameter. Some subspace \(G_{p)}(-\pi, \pi)\) of the space \(L_{p)}(-\pi, \pi)\) and the grand-Hardy classes \(H_{p)}^+\) and \(_mH_{p)}^-\) are considered. Using the analog of Cauchy representation theorem in grand-Hardy classes, the subspaces \(GH_{p)}^+\) and \(_mGH_{p)}^-\) are defined and the solvability of nonhomogeneous Riemann problem in the classes \(GH_{p)}^+\times_mGH_{p)}^-\) is studied. The subspaces \(G_{p)}^+(-\pi, \pi)\) and \(G_{p)}^-(-\pi, \pi)\) of the space \(G_{p)}(-\pi, \pi)\) are defined and the basicity of the systems \(\left\{e^{int}\right\}_{n\in \mathbb Z_+}\) and \(\left\{e^{-int}\right\}_{n\in \mathbb N}\) for these subspaces, respectively, is established. Finally, using boundary value problems method, the basicity of the system \(\left\{e^{i(n - \beta \operatorname{sign}n)t}\right\}_{n\in \mathbb Z}\) for \(G_{p)}(-\pi, \pi)\) is established.Finite decomposition of Herz-type Hardy spaces for the Dunkl operatorhttps://zbmath.org/1528.460292024-03-13T18:33:02.981707Z"Lachiheb, Mehdi"https://zbmath.org/authors/?q=ai:lachiheb.mehdiSummary: The corresponding Herz-type Hardy spaces to new weighted Herz spaces \(HK^{\beta, p}_{\alpha, q}\) associated with the Dunkl operator on \(\mathbb{R}\) have been characterized by atomic decompositions. Later a new characterization of \(HK^{\beta, p}_{\alpha, q}\) on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl operator.Non-compact embeddings of Sobolev spaceshttps://zbmath.org/1528.460302024-03-13T18:33:02.981707Z"Edmunds, D. E."https://zbmath.org/authors/?q=ai:edmunds.david-eric"Lang, J."https://zbmath.org/authors/?q=ai:lang.janSummary: The paper deals with the embedding of a first-order Sobolev space in a Lebesgue space on a bounded open set, and gives an example in which the embedding is not strictly singular. The same is shown to hold when the open set is an infinite strip.Density results and trace operator in weighted Sobolev spaces defined on the half-line, equipped with power weightshttps://zbmath.org/1528.460312024-03-13T18:33:02.981707Z"Kaczmarek, Radosław"https://zbmath.org/authors/?q=ai:kaczmarek.radoslaw"Kałamajska, Agnieszka"https://zbmath.org/authors/?q=ai:kalamajska.agnieszkaSummary: We study properties of \(W_0^{1, p} (\mathbb{R}_+, t^\beta)\) -- the completion of \(C_0^\infty (\mathbb{R}_+)\) in the power-weighted Sobolev spaces \(W^{1, p} (\mathbb{R}_+, t^\beta)\), where \(\beta \in \mathbb{R}\). Among other results, we obtain the analytic characterization of \(W_0^{1, p} (\mathbb{R}_+, t^\beta)\) for all \(\beta \in \mathbb{R}\). Our analysis is based on the precise study of the two trace operators: \(Tr^0 (u) := \lim_{t \to 0} u (t)\) and \(Tr^\infty (u) := \lim_{t \to \infty} u (t)\), which leads to the analysis of the asymptotic behavior of functions from \(W_0^{1, p} (\mathbb{R}_+, t^\beta)\) near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODEs, or PDEs with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by \textit{M. Cwikel} and \textit{A. Einav} [J. Funct. Anal. 277, No. 7, 2381--2441 (2019; Zbl 1435.46017)], which we discuss within the particular case of Sobolev spaces \(W^{1, p} (\mathbb{R}_+, t^\beta)\).Sharp second-order Adams inequalities in Lorentz-Sobolev spaceshttps://zbmath.org/1528.460322024-03-13T18:33:02.981707Z"Tang, Hanli"https://zbmath.org/authors/?q=ai:tang.hanliAuthors' abstract: In this paper, we establish sharp subcritical and critical second-order Adams inequalities in Lorentz-Sobolev spaces. We also prove the subcritical and critical Adams inequalities are actually equivalent and our results extend existing ones in [the author, Potential Anal. 53, No.~1, 297--314 (2020; Zbl 1456.46034)] to second order.
Reviewer: Şerban Costea (Piteşti)Complementations in \(C(K,X)\) and \(\ell_\infty (X)\)https://zbmath.org/1528.460332024-03-13T18:33:02.981707Z"Candido, Leandro"https://zbmath.org/authors/?q=ai:candido.leandroLet \(X\) be a Banach space and let \(K\) be a compact Hausdorff topological space. Denote by \(C(K,X)\) the Banach space of continuous \(X\)-valued functions defined on \(K\) endowed with the supremum norm and denote by \(\ell_{\infty}(X)\) the Banach space of all bounded sequences in \(X\), also endowed with the supremum norm. The first part of this paper is devoted, in the author's own words, ``to present another Cembranos-Freniche type theorem'' (on complemented copies of \(c_0\) in \(C(K,X)\)). The second part deals with the following general question and some related problems: given two compact Hausdorff topological spaces, \(K_1\) and \(K_2\), such that \(\ell_{\infty}(C(K_1))\) and \(\ell_{\infty}(C(K_2))\) are isomorphic, what properties must \(K_1\) and \(K_2\) share?
The article also provides a good, brief survey of the research field and contains a very complete list of references. So people interested in this kind of problems or in these spaces should know it.
Reviewer: José Mendoza (Madrid)Topologies on the set of iterates of a holomorphic function in infinite dimensionshttps://zbmath.org/1528.460342024-03-13T18:33:02.981707Z"Mackey, M."https://zbmath.org/authors/?q=ai:mackey.michael-t"Mellon, P."https://zbmath.org/authors/?q=ai:mellon.paulineLet \(E\) be a complex Banach space with open unit ball \(B\) and let \(f:B \to B\) be a holomorphic mapping such that \(f(B)\) is a relatively compact set. The subject of the present paper is the sequence of iterates \((f^n)_n\) as a subset of \(H(B;E)\) endowed with either the pointwise topology, the compact-open topology or the local uniform topology \(\tau,\) that is, the topology of uniform convergence on some neighbourhood of every \(x\in B.\)
Among other results, the following stand out:
\begin{itemize}
\item For any subsequence \( (f^{n_k})_k\) and any complex extreme point \(e\in \overline{B},\) the following are equivalent:
(a) \(\lim_k f^{n_k}(z)_k=e\) for some \(z\in B;\)
(b) \(\lim_k f^{n_k}(z)_k=e\) for all \(z\in B;\)
(c) \( (f^{n_k})_k\) is \(\tau\)-convergent to the constant function \(e.\)
\item The pointwise limit of any convergent subsequence \( (f^{n_k})_k\) is a holomorphic mapping.
\item A subnet \( (f^{n_\alpha})_\alpha\) is pointwise convergent if, and only if, it is \(\tau\)-convergent.
\end{itemize}
Reviewer: Pablo Galindo (València)Analytic Fourier-Feynman transforms associated with bounded linear operators on abstract Wiener spaceshttps://zbmath.org/1528.460352024-03-13T18:33:02.981707Z"Choi, Jae Gil"https://zbmath.org/authors/?q=ai:choi.jae-gilSummary: In this paper, we introduce the concepts of the analytic Feynman integral and the analytic Fourier-Feynman transform associated with bounded linear operators on abstract Wiener spaces. We then investigate Fubini theorems for the analytic Feynman integrals and the transforms. The Fubini theorems for the transforms investigated in this paper are to express the iterated Fourier-Feynman transform associated with bounded linear operators as a single Fourier-Feynman transform.Weighted holomorphic mappings attaining their normshttps://zbmath.org/1528.460362024-03-13T18:33:02.981707Z"Jiménez-Vargas, A."https://zbmath.org/authors/?q=ai:jimenez-vargas.antonioGiven an open set \(U \subseteq \mathbb{C}^n\), a complex Banach space \(F\), and a weight \(v: U \rightarrow \mathbb{R}_{\geq 0}\), the author investigates the space of weighted holomorphic mappings \(\mathcal{H}_v(U, F)\). He focuses on studying weighted holomorphic mappings that attain their weighted supremum norms, that is, those \(f \in \mathcal{H}_v(U, F)\) for which there exists \(z \in U\) satisfying
\[
\|f(z)\|=\|f\|_v:= \sup \{ v(u) \|f(u)\| :u\in U\}.
\]
Let \(\mathcal{H}_{v_0}(U, F) \subset \mathcal{H}_v(U, F)\) denote the set of functions \(g\) such that \(vg\) vanishes at infinity. In this work, the author shows that, if the closed unit ball of \(\mathcal{H}_{v_0}(U, F)\) is compact-open dense on the unit ball of \(\mathcal{H}{v}(U, F)\), then the set of norm-dense weighted holomorphic mappings that achieve their weighted supremum norms is dense.
Reviewer: Jorge Tomás Rodríguez (Tandil)Approximate amenability and pseudo-amenability in Banach algebrashttps://zbmath.org/1528.460372024-03-13T18:33:02.981707Z"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.59|zhang.yong.13|zhang.yong.28|zhang.yong.21|zhang.yong.5|zhang.yong.15|zhang.yong.19|zhang.yong.4|zhang.yong.9|zhang.yong.18|zhang.yong.14|zhang.yong|zhang.yong.10|zhang.yong.2|zhang.yong.8|zhang.yong.41|zhang.yong.12Let \(A\) be a Banach algebra and \(X\) a Banach \(A\)-bimodule. A linear mapping \(D: A\to X\) is a derivation if
\[
D(ab) = a . D(b) + D(a) . b\quad (a, b \in A),
\]
where \(.\) denotes the module action of \(A\) on \(X\). In particular, for each \(\xi\in X,\) the mapping \(a \mapsto \delta_\xi(a) := a.\xi-\xi.a\) is a derivation, which is called an inner derivation. The derivation \(D\) is called (boundedly) approximately inner if there is a (resp. bounded) net \((\xi_\alpha) \subseteq X\) such that
\[
D(a) = \lim_\alpha (a\xi_\alpha-\xi_\alpha a), \quad (a\in A).
\]
A (bounded) net \((u_i)\) in the projective tensor product \(A\hat{\otimes} A\) is a (resp. bounded) approximate diagonal for $A$ if
\[
\lim_i (a . u_i - u_i . a) = 0,\quad \lim_i\pi(u_i)a \to a \quad (a\in A),
\]
where \(\pi: A\hat{\otimes}A \to A\) is the product mapping defined by \(\pi(a \otimes b) = ab.\)
The Banach algebra \(A\) is amenable if for each Banach \(A\)-bimodule \(X\), every continuous derivation \(D: A\to X\) is boundedly approximately inner [\textit{F.~Gourdeau}, Math. Proc. Camb. Philos. Soc. 112, No.~3, 581--588 (1992; Zbl 0782.46043)]. This definition is equivalent to ``for each Banach \(A\)-bimodule \(X\), every continuous derivation from \(A\) into the dual module \(X^*\) is inner'' as defined in [\textit{B.~E. Johnson}, Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)], and also equivalent to ``there is a bounded approximate diagonal for $A$'', see [\textit{B.~E. Johnson}, Am. J. Math. 94, 685--698 (1972; Zbl 0246.46040)]. The amenability of Banach algebras may be generalized by dropping the above boundedness conditions as follows.
The Banach algebra \(A\) is approximately amenable if for each Banach \(A\)-bimodule \(X\), every continuous derivation \(D: A \to X\) is approximately inner [\textit{F.~Ghahramani} and \textit{R.~J. Loy}, J. Funct. Anal. 208, No.~1, 229--260 (2004; Zbl 1045.46029)]; equivalently, the Banach algebra \(A\) is approximately amenable if for each Banach \(A\)-bimodule \(X\), every continuous derivation from \(A\) to \(X^*\) is approximately inner, see [\textit{F.~Ghahramani} et al., J. Funct. Anal. 254, No.~7, 1776--1810 (2008; Zbl 1146.46023)].
Also, the Banach algebra \(A\) is called pseudo-amenable if there exists an approximate diagonal for \(A\).
Approximate amenability and pseudo-amenability are different notions, in general.
In this paper, the recent development of approximate amenability and pseudo-amenability in Banach algebras, concentrating on the relationship between the two notions is surveyed. It has been conjectured that a Banach algebra with a bounded approximate identity is approximately amenable if and only if it is pseudo-amenable. Some partial affirmative results concerning the conjecture are given.
Reviewer: Sedigheh Barootkoob (Bojnūrd)Some applications of simultaneous continuous functional calculushttps://zbmath.org/1528.460382024-03-13T18:33:02.981707Z"Mazighi, M."https://zbmath.org/authors/?q=ai:mazighi.mohamed"El Kinani, A."https://zbmath.org/authors/?q=ai:el-kinani.abdellahThe notion of a continuous functional calculus is well known in the framework of a \(C^\ast\)-algebra \(A\) (unital or not) and for a normal element \(a \in A\) (\(a^\ast a = aa^\ast\)). This notion, along with the Gel'fand transform, is the essence of reducing several abstract problems concerning certain elements to problems in the function algebra context. The respective notion of a continuous operational calculus has also been given for a finite commutative family of normal bounded operators on a Hilbert space.
In the present paper, the authors extend the latter notion, by defining a simultaneous continuous functional calculus in a \(C^\ast\)-algebra \(A\) with unit \(e\) and for a commutative family \({\mathbf{a}}=(a_i)_{i\in I}\) of normal elements of \(A\). For this, they employ a generalized version of the Gel'fand transform, say \(\hat{\mathbf{a}}\), defined by \(\hat{{\mathbf{a}}}(\chi) = (\chi(a_i))_{i \in I} \in \mathbb{C}^I, \) for every \(\chi \) in the Gel'fand spectrum \(Sp(B)\) (viz. the set of all non-zero characters of the closed subalgebra \( B\) generated by \(\mathbf{a}\) and \(e\), being actually a unital commutative \(C^\ast\)-algebra). In that case, the continuous functional calculus is a unique unitary \(^\ast \)-morphism, say \(\boldsymbol{\Phi}_{\mathbf{a}} \), of the \(C^\ast\)-algebra \(C( Sp(\mathbf{a})) \) (of all continuous complex-valued functions on the simultanuous spectrum of \(\mathbf{a}\) (viz. \(Sp({\mathbf{a}})=\hat{\mathbf{a}}(Sp(B))\)), into \(A\), satisfying the relation
\[
\hat{\boldsymbol{\Phi}}_{\mathbf{a}}(f)=f \circ \mathbf{\hat{a}}, \text { for every} \, \ f \in C( Sp(\mathbf{a}))\tag{\(\ast\)}.
\]
\(\boldsymbol{\Phi}_{\mathbf{a}} \) is isometric as well and its image is the sub-\(C^\ast \)-algebra of \(A\) generated by \({\mathbf{a}}\) and \(e\). By employing \( ( \ast ) \), the authors state the analogous (simultaneous) spectral mapping theorem. Moreover, they apply the simultaneous continuous calculus and get an orthonormal basis on a certain locally Hilbert space \(H\) and for a commutative family of normal operators on \(H\). Further, the authors are interested in the convergence of Fourier series, as is stated by Norbert Wiener (Wiener's lemma) and later by Paul Lévy in a more general form, known as the Wiener-Lévy theorem. So, they provide analogous results, as applications of their present work, where also a certain regular weight is employed.
Reviewer: Marina Haralampidou (Athína)Quasitriangular operator algebrashttps://zbmath.org/1528.460392024-03-13T18:33:02.981707Z"Amini, Massoud"https://zbmath.org/authors/?q=ai:amini.massoud"Moradi, Mehdi"https://zbmath.org/authors/?q=ai:moradi.mehdi"Mousavi, Ismaeil"https://zbmath.org/authors/?q=ai:mousavi.ismaeilSummary: We give characterizations of quasitriangular operator algebras along the line of Voiculescu's characterization of quasidiagonal \(C^\ast\)-algebras.\(R^\ast\)-algebras and \(C^\ast\)-uniqueness of pre-\(C^\ast\)-algebrashttps://zbmath.org/1528.460402024-03-13T18:33:02.981707Z"An, Guimei"https://zbmath.org/authors/?q=ai:an.guimei"Gao, Mingchu"https://zbmath.org/authors/?q=ai:gao.mingchuSummary: In 1967, Kadison raised the following question [\textit{L. M. Ge}, Acta Math. Sin., Engl. Ser. 19, No. 3, 619--624 (2003; Zbl 1043.46045)]: if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In [``On regular $^*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras'', Preprint (2021), \url{arXiv:2107.05806}], \textit{M.~Mori} connected Kadison's question to von Neumann's concept of regular rings from the 1930s, and introduced a notion of \(R^\ast\)-algebras.
In this paper, we study \(R^\ast\)-algebras, the Kadison's question, and a related question of uniqueness of \(C^\ast\)-norm on pre-\(C^\ast\)-algebras. Precisely, we study atomic \(R^\ast\)-algebras, and find a maximal ultramatricial \(R^\ast\)-subalgebra of an atomic \(R^\ast\)-algebra, which properly contains the maximal purely atomic \(R^\ast\)-subalgebra generated by all finite projections of the algebra. We prove that the set of all \(\ast\)-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of \(\ast\)-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product \(\ast\)-algebra \(\mathcal{B}\rtimes_\alpha G\), where \(G\) is a locally finite group and \(\mathcal{B}\) is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra.
We characterize uniqueness of \(C^\ast\)-norm on a pre-\(C^\ast\)-algebras in terms of its \(\ast\)-representations, weak containment of its \(\ast\)-representations, ideals of its enveloping \(C^\ast\)-algebra, or the primitive ideal space of the \(C^\ast\)-algebra. We also prove that a unital pre-\(C^\ast\)-algebra has a unique \(C^\ast\)-norm if and only if, for a faithful \(\ast\)-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the \(\ast\)-representation. These characterizations provide answers to the question raised by M. Mori [loc.\,cit.]\ on finding a sufficient and necessary condition for a \(\ast\)-algebra to have a unique \(C^\ast\)-norm. For discrete groups, we give a characterization for a group algebra to have a unique \(C^\ast\)-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing \(C^\ast\)-unique discrete groups
[\textit{V.~Alekseev}, ``(Non)-uniqueness of $C^*$-norms on group rings of amenable groups'', pp.~2292--2293 in: \textit{M.~Rørdam} (ed.) et al., Oberwolfach Rep. 16, No.~3, 2257--2332 (2019; Zbl 1450.00018)]The Calkin algebra is not countably homogeneoushttps://zbmath.org/1528.460412024-03-13T18:33:02.981707Z"Farah, Ilijas"https://zbmath.org/authors/?q=ai:farah.ilijas"Hirshberg, Ilan"https://zbmath.org/authors/?q=ai:hirshberg.ilanSummary: We show that the Calkin algebra is not countably homogeneous, in the sense of continuous model theory. We furthermore show that the connected component of the unitary group of the Calkin algebra is not countably homogeneous.Irreducibility and monicity for representations of \(k\)-graph \(C^*\)-algebrashttps://zbmath.org/1528.460422024-03-13T18:33:02.981707Z"Farsi, Carla"https://zbmath.org/authors/?q=ai:farsi.carla"Gillaspy, Elizabeth"https://zbmath.org/authors/?q=ai:gillaspy.elizabeth"Goncalves, Daniel"https://zbmath.org/authors/?q=ai:goncalves.danielSummary: The representations of a \(k\)-graph \(C^*\)-algebra \(C^*(\Lambda)\) which arise from \(\Lambda\)-semibranching function systems are closely linked to the dynamics of the \(k\)-graph \(\Lambda\). In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of \(\Lambda\); our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible \(\Lambda\)-semibranching representations and purely atomic representations of \(C^*(\Lambda)\). Throughout the paper, we work in the setting of row-finite source-free \(k\)-graphs; this paper constitutes the first analysis of \(\Lambda\)-semibranching representations at this level of generality.Norm inequalities associated with two projectionshttps://zbmath.org/1528.460432024-03-13T18:33:02.981707Z"Tian, Xiaoyi"https://zbmath.org/authors/?q=ai:tian.xiaoyi"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiang"Zhang, Xiaofeng"https://zbmath.org/authors/?q=ai:zhang.xiaofengSummary: Suppose that \(p\) and \(q\) are projections in a unital \(C^*\)-algebra \(\mathfrak{A}\) such that \(\Vert p(1-q)\Vert <1\). It is shown that there exists a unitary \(u\) in \(\mathfrak{A}\) which is homotopic to the unit of \(\mathfrak{A} \), and satisfies \(pup=pu^*p\), \(u(pqp)u^*=qpq\) and
\[
\Vert 1-u\Vert \le \sqrt{\frac{2\Vert (qp)^\dag \Vert }{1+\Vert (qp)^\dag \Vert }}\cdot \Vert p(1-q)\Vert,
\]
where \((qp)^\dag\) denotes the Moore-Penrose inverse of \(qp\). Under the same restriction of \(\Vert p(1-q)\Vert <1\), it is proved that \(\Vert p-q\Vert <1\) if and only if there exists a unitary \(u\) in \(\mathfrak{A}\) such that \(pup\) is normal and \(q=upu^*\). An example is constructed to show that there exist certain Hilbert space \(H\) and projections \(p\) and \(q\) on \(H\) such that \(\Vert p-q\Vert =1\) and \(q=upu^*\) for some unitary operator \(u\) on \(H\).Distance between unitary orbits of self-adjoint elements in \(C^*\)-algebras of tracial rank onehttps://zbmath.org/1528.460442024-03-13T18:33:02.981707Z"Wang, Ruofei"https://zbmath.org/authors/?q=ai:wang.ruofeiSummary: The note studies certain distance between unitary orbits. A result about Riesz interpolation property is proved in the first place. \textit{H. Weyl} [Math. Ann. 71, 441--479 (1912; JFM 43.0436.01)] showed that \(\mathrm{dist}(U(x), U(y)) = \delta (x, y)\) for self-adjoint elements in matrices. The author generalizes the result to \(C^*\)-algebras of tracial rank one. It is proved that dist \(\mathrm{dist}(U(x), U(y)) = D_c (x, y)\) in unital \textit{AT}-algebras and in unital simple \(C^*\)-algebras of tracial rank one, where \(x, y\) are self-adjoint elements and \(D_C (x, y)\) is a notion generalized from \(\delta (x, y)\).Analytic continuation of concrete realizations and the McCarthy Champagne conjecturehttps://zbmath.org/1528.460452024-03-13T18:33:02.981707Z"Bickel, Kelly"https://zbmath.org/authors/?q=ai:bickel.kelly"Pascoe, J. E."https://zbmath.org/authors/?q=ai:pascoe.james-eldred"Tully-Doyle, Ryan"https://zbmath.org/authors/?q=ai:tully-doyle.ryanSummary: In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz, and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges-Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions that are rational in one of the variables (so-called \textit{quasi-rational functions}). We then establish a positive solution to McCarthy's Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and \(d\)-variable perspective functions.Spectral bounds for the quantum chromatic number of quantum graphshttps://zbmath.org/1528.460462024-03-13T18:33:02.981707Z"Ganesan, Priyanga"https://zbmath.org/authors/?q=ai:ganesan.priyangaSummary: Quantum graphs are an operator space generalization of classical graphs that have emerged in different branches of mathematics including operator theory, non-commutative topology and quantum information theory. In this paper, we obtain lower bounds for the classical, quantum and quantum-commuting chromatic number of a quantum graph using eigenvalues of the quantum adjacency matrix. In particular, we prove a quantum generalization of Hoffman's bound and introduce quantum analogues for the edge number, Laplacian and signless Laplacian. We generalize all the spectral bounds of \textit{C.~Elphick} and \textit{P.~Wocjan} [J. Comb. Theory, Ser. A 168, 338--347 (2019; Zbl 1421.05042)]
to the quantum graph setting and demonstrate the tightness of these bounds in the case of complete quantum graphs. Our results are achieved using techniques from linear algebra and a combinatorial definition of quantum graph coloring, which is obtained from the winning strategies of a quantum-to-classical nonlocal graph coloring game [\textit{M.~Brannan} et al., J. Math. Phys. 63, No.~11, Article ID 112204, 34~p. (2022; Zbl 1508.91108)].\(C^*\)-isomorphisms associated with two projections on a Hilbert \(C^*\)-modulehttps://zbmath.org/1528.460472024-03-13T18:33:02.981707Z"Fu, Chunhong"https://zbmath.org/authors/?q=ai:fu.chunhong"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiang"Yan, Guanjie"https://zbmath.org/authors/?q=ai:yan.guanjieSummary: Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies \(C^*\)-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple \((P, Q, H)\) is said to be matched if \(H\) is a Hilbert \(C^*\)-module, \(P\) and \(Q\) are projections on \(H\) such that their infimum \(P \wedge Q\) exists as an element of \(\mathcal{L}(H)\), where \(\mathcal{L}(H)\) denotes the set of all adjointable operators on \(H\). The \(C^*\)-subalgebras of \(\mathcal{L}(H)\) generated by elements in \(\{P - P \wedge Q, Q - P \wedge Q, I\}\) and \(\{P, Q, P \wedge Q, I\}\) are denoted by \(i(P, Q, H)\) and \(o(P, Q, H)\), respectively. It is proved that each faithful representation \((\pi, X)\) of \(o(P, Q, H)\) can induce a faithful representation \((\widetilde{\pi}, X)\) of \(i(P, Q, H)\) such that
\[
\begin{aligned}
\widetilde{\pi} (P - P \wedge Q) = \pi (P) - \pi (P) \wedge \pi (Q),\\
\widetilde{\pi} (Q - P \wedge Q) = \pi (Q) - \pi (P) \wedge \pi (Q).
\end{aligned}
\]
When \((P, Q)\) is semi-harmonious, that is, \(\overline{\mathcal{R}(P + Q)}\) and \(\overline{\mathcal{R}(2I - P - Q)}\) are both orthogonally complemented in \(H\), it is shown that \(i(P, Q, H)\) and \(i(I - Q, I - P, H)\) are unitarily equivalent via a unitary operator in \(\mathcal{L}(H)\). A counterexample is constructed, which shows that the same may be not true when \((P, Q)\) fails to be semi-harmonious. Likewise, a counterexample is constructed such that \((P, Q)\) is semi-harmonious, whereas \((P, I - Q)\) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert \(C^*\)-modules are also provided.Thick elements and states in \(C^*\)-algebras in view of frame theoryhttps://zbmath.org/1528.460482024-03-13T18:33:02.981707Z"Fufaev, D. V."https://zbmath.org/authors/?q=ai:fufaev.d-vSummary: We study some classes of noncommutative \(C^*\)-algebras and generalize some results which were originally obtained for commutative algebras in topological terms. In particular, we are interested in results obtained for topological spaces with properties close to separability and \(\sigma\)-compactness. To obtain the algebraic, noncommutative versions of corresponding properties, we define and use the notions of thick elements and states. In particular, an element is thick if the only element orthogonal to it is zero.Comparison properties for asymptotically tracially approximation \(C^*\)-algebrashttps://zbmath.org/1528.460492024-03-13T18:33:02.981707Z"Fan, Qing Zhai"https://zbmath.org/authors/?q=ai:fan.qingzhai"Fang, Xiao Chun"https://zbmath.org/authors/?q=ai:fang.xiaochunSummary: We show that the following properties of the \(C^*\)-algebras in a class \(\mathcal{P}\) are inherited by simple unital \(C^*\)-algebras in the class of asymptotically tracially in \(\mathcal{P} \): (1) \(n\)-comparison, (2) \(\alpha \)-comparison (\(1 \leq \alpha < \infty \)).Hereditary uniform property \(\Gamma\)https://zbmath.org/1528.460502024-03-13T18:33:02.981707Z"Lin, Huaxin"https://zbmath.org/authors/?q=ai:lin.huaxinThe notion of property Gamma was introduced by Murray and von Neumann to show the existence of non-hyperfinite \(\mathrm{II}_1\) factors. A \(C^*\)-version of this notion known as uniform property Gamma has become an important regularity property for simple unital nuclear \(C^*\)-algebras [\textit{J.~Castillejos} et al., Invent. Math. 224, No.~1, 245--290 (2021; Zbl 1467.46055)]. It was shown in [\textit{J.~Castillejos} et al., Int. Math. Res. Not. 2022, No.~13, 9864--9908 (2022; Zbl 1506.46047)] that under the extra assumption of uniform property Gamma, the Toms-Winter regularity conjecture holds. The goal of the paper under review is twofold: extending uniform property Gamma to the setting of: (i) compact sets of quasitraces (in contrast to traces), (ii) non-unital simple separable \(C^*\)-algebras.
In the non-unital setting, the author introduces the notion of hereditary uniform property Gamma which generalises the previous unital notion. This differs from a previous generalisation to the non-unital case introduced in [\textit{J.~Castillejos} and \textit{S.~Evington}, Proc. Am. Math. Soc. 149, No.~11, 4725--4737 (2021; Zbl 1484.46066)]. An important result in this paper is that hereditary uniform property Gamma implies stable rank one. The main result establishes that the Toms-Winter regularity conjecture holds in the non-unital case under the extra assumption of hereditary uniform property Gamma.
Reviewer: Jorge Castillejos (Cuernavaca)Ergodicity of exclusion semigroups constructed from quantum Bernoulli noiseshttps://zbmath.org/1528.460512024-03-13T18:33:02.981707Z"Chen, Jinshu"https://zbmath.org/authors/?q=ai:chen.jinshu"Hai, Shexiang"https://zbmath.org/authors/?q=ai:hai.shexiangSummary: Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.Superconvergence and regularity of densities in free probabilityhttps://zbmath.org/1528.460522024-03-13T18:33:02.981707Z"Bercovici, Hari"https://zbmath.org/authors/?q=ai:bercovici.hari"Wang, Jiun-Chau"https://zbmath.org/authors/?q=ai:wang.jiun-chau"Zhong, Ping"https://zbmath.org/authors/?q=ai:zhong.pingSummary: The phenomenon of superconvergence, first observed in the central limit theorem of free probability, was subsequently extended to arbitrary limit laws for free additive convolution. We show that the same phenomenon occurs for the multiplicative versions of free convolution on the positive line and on the unit circle. We also show that a certain Hölder regularity, first demonstrated by \textit{P. Biane} [Indiana Univ. Math. J. 46, No. 3, 705--718 (1997; Zbl 0904.46045)] for the density of a free additive convolution with a semicircular law, extends to free (additive and multiplicative) convolutions with arbitrary freely infinitely divisible distributions.Wasserstein distance between noncommutative dynamical systemshttps://zbmath.org/1528.460532024-03-13T18:33:02.981707Z"Duvenhage, Rocco"https://zbmath.org/authors/?q=ai:duvenhage.roccoSummary: We introduce and study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein distance arises, but also study the asymmetric case. This setup is illustrated in the context of reduced dynamics, and a number of simple examples are also presented.\(C^*\)-algebras generated by multiplication operators and composition operators by functions with self-similar brancheshttps://zbmath.org/1528.460542024-03-13T18:33:02.981707Z"Hamada, Hiroyasu"https://zbmath.org/authors/?q=ai:hamada.hiroyasuLet \((K, d)\) be a compact metric space and let \(\gamma = (\gamma_1, \dots, \gamma_n)\) be a family of proper contractions of \(K\). We say that \(K\) is self-similar with respect to \(\gamma\) if \(K = \bigcup_{i=1}^n \gamma_i(K)\), and we say that \(\gamma\) satisfies the open set condition on \(K\) if there exists a nonempty open subset \(V\) of \(K\) such that \(\bigcup_{i=1}^n \gamma_i(V)\) is a subset of \(V\) and the \(\gamma_i(V)\) are pairwise disjoint.
Let \(\mathcal{B}(K)\) denote the \(\sigma\)-algebra of Borel subsets of \(K\). The Hutchinson measure is the unique probability measure \(\mu^H\) on \(\mathcal{B}(K)\) with the feature that \(\mu^H(E) = \sum_{i=1}^n\frac{1}{n}\mu^H(\gamma_i^{-1}(E))\) for every Borel subset \(E\) of \(K\). Suppose that \(\phi: K \rightarrow K\) is continuous and that the \(\gamma_i\) are inverse branches of \(\phi\); i.e., that \(\phi(\gamma_i(x)) = x\) for all \(x \in K\) and \(1 \leq i \leq n\). Consider the composition operator \(C_\phi\) on \(L^2(K, \mathcal{B}(K), \mu^H)\), and let \(\mathcal{MC}_\phi\) be the \(C^*\)-subalgebra of bounded operators on \(L^2(K, \mathcal{B}(K), \mu^H)\) generated by \(C_\phi\) and the set \(\{M_\psi: \psi \in C(K)\}\) of multiplication operators.
In this paper, the author shows that \(\mathcal{MC}_\phi\) can be realized as a Cuntz-Pimsner algebra. Specifically, define \(\mathcal{C} := \bigcup_{i=1}^n\{(\gamma_i(y), y) : y\in K\}\) and let \(Y = C(\mathcal{C})\). Then \(Y\) is a \(C(K)\operatorname{-}C(K)\) bimodule over \(C(K)\) in a natural way; the corresponding Cuntz-Pimsner algebra, which the author denotes \(\mathcal{O}_\gamma( K)\), is \(C^*\)-isomorphic to \(\mathcal{MC}_\phi\). The paper concludes with some examples.
Reviewer: Efton Park (Fort Worth)Amenability for actions of étale groupoids on \(C^*\)-algebras and Fell bundleshttps://zbmath.org/1528.460552024-03-13T18:33:02.981707Z"Kranz, Julian"https://zbmath.org/authors/?q=ai:kranz.julianSummary: We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable \(C^*\)-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the \(C^*\)-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid actions, we show that the approximation property implies measurewise amenability of the underlying action.Non-existence of translation-invariant derivations on algebras of measurable functionshttps://zbmath.org/1528.460562024-03-13T18:33:02.981707Z"Ber, Aleksey"https://zbmath.org/authors/?q=ai:ber.aleksey"Huang, Jinghao"https://zbmath.org/authors/?q=ai:huang.jinghao"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aSummary: Let \(S(0,1)\) be the \(*\)-algebra of all classes of Lebesgue measurable functions on the unit interval (0,1) and let \((\mathcal{A}, \| \cdot \|_{\mathcal{A}})\) be a complete symmetric \(\Delta\)-normed \(*\)-subalgebra of \(S(0,1)\), in which simple functions are dense, e.g., \(L_\infty (0,1)\), \(L_{\log}(0,1)\), \(S(0,1)\) and the Arens algebra \(L^\omega (0,1)\) equipped with their natural \(\Delta\)-norms. We show that there exists no non-trivial derivation \(\delta : \mathcal{A} \to S(0,1)\) commuting with all dyadic translations of the unit interval. Let \(\mathcal{M}\) be a type \textrm{II} (or I\(_\infty)\) von Neumann algebra, \(\mathcal{A}\) be an arbitrary abelian von Neumann subalgebra of \(\mathcal{M}\), let \(S(\mathcal{M})\) be the algebra of all measurable operators affiliated with \(\mathcal{M}\). We show that there exists no non-trivial derivation \(\delta : \mathcal{A} \to S(A)\) which admits an extension to a derivation on \(S(\mathcal{M})\). In particular, we answer an untreated question in [\textit{A.~Ber} et al., J. Funct. Anal. 279, No.~5, Article ID 108589, 25~p. (2020; Zbl 1461.46056)].Decomposable product systems associated to non-stationary Poisson processeshttps://zbmath.org/1528.460572024-03-13T18:33:02.981707Z"Shanmugasundaram, Sundar"https://zbmath.org/authors/?q=ai:shanmugasundaram.sundarSummary: Let \(P\) be a closed convex cone in \(\mathbb{R}^d\), which we assume is pointed and spanning, i.e., \(P\cap(-P)=\{0\}\) and \(P-P=\mathbb{R}^d\). We demonstrate that, when \(d\geq 2\), in contrast to the one-parameter situation, Poisson processes on \(\mathbb{R}^d\), with intensity measure absolutely continuous with respect to the Lebesgue measure, restricted to \(P\)-invariant closed subsets, provide us with a source of examples of decomposable \(E_0\)-semigroups that are not always CCR flows.Frobenius \(C^\ast\)-algebras and local adjunctions of \(C^\ast\)-correspondenceshttps://zbmath.org/1528.460582024-03-13T18:33:02.981707Z"Crisp, Tyrone"https://zbmath.org/authors/?q=ai:crisp.tyroneSummary: Generalizing the well-known correspondence between two-sided adjunctions and Frobenius algebras, we establish a one-to-one correspondence between local adjunctions of \(C^\ast\)-correspondences, as defined and studied in prior work with \textit{P.~Clare} and \textit{N.~Higson} [J. Inst. Math. Jussieu 17, No.~2, 453--488 (2018; Zbl 1395.46043)], and Frobenius \(C^\ast\)-algebras, a~natural \(C^\ast\)-algebraic adaptation of the standard notion of Frobenius algebras that we introduce here.Group actions on twisted sums of Banach spaceshttps://zbmath.org/1528.460592024-03-13T18:33:02.981707Z"Castillo, Jesús M. F."https://zbmath.org/authors/?q=ai:castillo.jesus-m-f"Ferenczi, Valentin"https://zbmath.org/authors/?q=ai:ferenczi.valentinSummary: We study bounded actions of groups and semigroups \(G\) on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of \(G\)-centralizer and \(G\)-equivariant map. We will show that when (A) \(G\) is an amenable group and (U) the target space is complemented in its bidual by a \(G\)-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of \(G\)-centralizers; and that, under (A) and (U), \(G\)-centralizers are bounded perturbations of \(G\)-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of \(L_p(0,1)\), \(p\ne 2\) on the Kalton-Peck space \(Z_p\), certain non-unitarizable triangular representations of the free group \({\mathbb{F}}_\infty\) on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of \(L_p\)-spaces. In the penultimate section we consider the category of \(G\)-Banach spaces and study its exact sequences, showing that, under (A) and (U), \(G\)-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.The Rochberg gardenhttps://zbmath.org/1528.460602024-03-13T18:33:02.981707Z"Castillo, Jesús M. F."https://zbmath.org/authors/?q=ai:castillo.jesus-m-f"Pino, Raúl"https://zbmath.org/authors/?q=ai:pino.raulSummary: In 1996, it was published the seminal work of \textit{R.~Rochberg} ``Higher order estimates in complex interpolation theory'' [Pac. J. Math. 174, No.~1, 247--267 (1996; Zbl 0866.46047)]. Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each \(z \in \mathbb{S}\) the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation Theory, Homology, Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significant open problems.Local error bound property for nonlinear split feasibility problems in Banach spaceshttps://zbmath.org/1528.460612024-03-13T18:33:02.981707Z"Gao, Tianming"https://zbmath.org/authors/?q=ai:gao.tianming"Shen, Weiping"https://zbmath.org/authors/?q=ai:shen.weiping"Wang, Jinhua"https://zbmath.org/authors/?q=ai:wang.jinhua.1Summary: We study local error bound property for nonlinear split feasibility problems in Banach spaces. In terms of regular and quasiregular conditions, we provide some suitable sufficient conditions ensuring local error bound property for nonlinear split feasibility problems in Banach spaces.Clarke subdifferential for Lipschitz functions on Asplund spaceshttps://zbmath.org/1528.490092024-03-13T18:33:02.981707Z"Zheng, Xi Yin"https://zbmath.org/authors/?q=ai:zheng.xiyin|zheng.xi-yinThe author considers an Asplund space \(X\), an open and nonempty subset \(G\) of \(X\), and a locally Lipschitz function \(f:G\rightarrow \mathbb{R}\). \(X\) is an Asplund space if every continuous and convex function on an open and convex subset \(G\) of \(X\) is Fréchet differentiable on a dense subset \( \mathcal{G}_{F}(f)\) of \(G\). The first main result proves that, for any \(x\in G\), the Clarke directional derivative \(f^{o}(x,h)\) of \(f\) at \(x\) in the direction \(h\) defined through \(f^{o}(x,h)=\lim \sup_{y\rightarrow x,t\rightarrow 0^{+}}\frac{f(y+th)-f(y)}{t}\) is equal to: \(f^{o}(x,h)=\limsup_{y\overset{\mathcal{G}_{f}}{\rightarrow }x}\left\langle \nabla f(y),h\right\rangle \), \(\forall h\in X\) and the subdifferential \(\partial f(x)\) of \(f\) at \(x\) defined through \(\partial f(x)=\{x^{\ast }\in X^{\ast }:\left\langle x^{\ast },h\right\rangle \leq f^{o}(x,h)\), \(\forall x\in X\}\) is equal to: \(\partial f(x)=\overline{co} ^{w^{\ast }}(w^{\ast }-\lim_{k\rightarrow \infty }\nabla f(x_{k}):x_{k}\rightarrow x\), \(x_{k}\in \mathcal{G}_{F}(f)\}\), where \(y \overset{\mathcal{G}_{f}}{\rightarrow }x\) means \(\left\Vert y-x\right\Vert \rightarrow 0\) and \(y\in \mathcal{G}_{F}(f)\). For the proof, the author recalls that the unit ball \(B_{X}^{\ast }\) of the dual space \(X^{\ast }\) of an Asplund space \(X\) is sequentially compact with respect to the weak\(^{\ast }\) topology. The second main result proves that if \(X\) is a normed space, \( F:X\rightarrow \mathbb{R}^{n}\) is a piecewise linear vector valued function, and \(g:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is a locally Lipschitz function, then, for any \(x\in X\), the Jacobian chain rule \(\partial (g\circ F)(x)\subset co(\partial g(F(x))\partial F(x))\) holds true.
Reviewer: Alain Brillard (Riedisheim)Caristi-type conditions in constraint minimisation of mappings in metric and partially ordered spaceshttps://zbmath.org/1528.490132024-03-13T18:33:02.981707Z"Zhukovskiy, Evgeny"https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-s"Burlakov, Evgenii"https://zbmath.org/authors/?q=ai:burlakov.evgenii"Malkov, Ivan"https://zbmath.org/authors/?q=ai:malkov.ivan-nikolaevichSummary: We consider the problem of finding minima of mappings defined on metric and partially ordered spaces subject to constraints in the form of inclusions (and as a consequence in the form of equalities and/or inequalities). We introduce analogues of Caristi-type inequality proposed in the studies on minimisation of non-convex functionals in metric spaces. Statements on attainment of minima of non-convex functionals on the solutions of the corresponding inclusions are proved. The proofs are based on the construction of a mapping for which the point of unconstraint minimum is the sought-for constraint minimum for the problem under consideration. We provide conditions for stability of the constraint minima to perturbations of the minimised functionals and the constraints. We also establish connections between the obtained statements on constraint minima, namely, we demonstrate that the statements on constraint minima of functionals in partially ordered spaces are more general than the corresponding statements for functionals defined on metric spaces.Geometric valuation theoryhttps://zbmath.org/1528.520082024-03-13T18:33:02.981707Z"Ludwig, Monika"https://zbmath.org/authors/?q=ai:ludwig.monikaRecall that a valuation on a family \(\mathcal{S}\) of subsets of euclidean space \(\mathbb{R}^n\) is a function \(Z\colon \mathcal{S} \to \mathbb{A}\), with \(\mathbb{A}\) an abelian semigroup, such that
\[
Z(K \cup L) + Z(K \cap L) = Z(K) + Z(L)
\]
whenever \(K,L \in \mathcal{S}\) are such that \(K \cup L, K \cap L \in \mathcal{S}\) also. In this paper, the author surveys (without proofs) results about valuations on convex bodies (including polytopes) and function spaces, which satisfy continuity conditions and invariance or covariance properties under certain groups acting on \(\mathbb{R}^n\). The groups in question are the translations and either the special linear group \(\mathrm{SL}(n)\) or special orthogonal group \(\mathrm{SO}(n)\), and apart from the scalars, the semigroups \(\mathbb{A}\) can be tensors or even convex bodies themselves.
This comprehensive survey covers results from the classical up to the very recent (we do not pick out individual topics, because once started it would be hard to know when to stop), and has a valuable bibliography of 139 items. It is much to be recommended to anyone who wishes to know the current state of affairs of the subject.
For the entire collection see [Zbl 1519.00033].
Reviewer: Peter McMullen (London)Directional derivatives for set-valued maps based on set convergenceshttps://zbmath.org/1528.540072024-03-13T18:33:02.981707Z"Durea, Marius"https://zbmath.org/authors/?q=ai:durea.mariusSummary: We explore the possibility to define and to meaningfully apply some new concepts of directional derivative which incorporate in their construction set convergences to set-optimization problems. We connect these new constructions with other directional derivatives for set-valued maps and we emphasize the flexibility and the potential applicability of this new approach. In this vein, we indicate a possible axiomatic perspective that allows one to significantly increase the number and (maybe) the efficiency of these derivatives when applied to concrete problems.Vector fields and flows on subcartesian spaceshttps://zbmath.org/1528.580052024-03-13T18:33:02.981707Z"Karshon, Yael"https://zbmath.org/authors/?q=ai:karshon.yael"Lerman, Eugene"https://zbmath.org/authors/?q=ai:lerman.eugene-mIn this paper, the authors investigate vector fields and flows on subcartesian spaces. Concretely, vector fields and the associated flows on a class of singular spaces are analyzed.
Reviewer: Savin Treanţă (Bucureşti)Schrödinger quantization of infinite-dimensional Hamiltonian systems with a nonquadratic Hamiltonian functionhttps://zbmath.org/1528.580112024-03-13T18:33:02.981707Z"Smolyanov, O. G."https://zbmath.org/authors/?q=ai:smolyanov.oleg-georgievich"Shamarov, N. N."https://zbmath.org/authors/?q=ai:shamarov.nikolai-nSummary: According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrödinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by \textit{R. P. Feynman} [Rev. Mod. Phys. 20, No. 2, 367--387 (1948; Zbl 1371.81126)]. In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.Contextual unification of classical and quantum physicshttps://zbmath.org/1528.810192024-03-13T18:33:02.981707Z"Van Den Bossche, Mathias"https://zbmath.org/authors/?q=ai:van-den-bossche.mathias"Grangier, Philippe"https://zbmath.org/authors/?q=ai:grangier.philippeSummary: Following an article by \textit{J. von Neumann} [Compos. Math. 6, 1--77 (1938; JFM 64.0377.01)] on infinite tensor products, we develop the idea that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorisation. By interpreting physically this mathematical fact, we show that it provides a natural way to describe the ``Heisenberg cut'', as well as a unified mathematical model including both quantum and classical physics, appearing as required incommensurable facets in the description of nature.Mentor initiated controlled bi-directional remote state preparation scheme for \((2 \iff 4)\)-qubit entangled states in noisy channelhttps://zbmath.org/1528.810362024-03-13T18:33:02.981707Z"Choudhury, Binayak S."https://zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Mandal, Manoj Kumar"https://zbmath.org/authors/?q=ai:mandal.manoj-kumar"Samanta, Soumen"https://zbmath.org/authors/?q=ai:samanta.soumenSummary: In this paper we present a bi-directional protocol for mutual remote preparation of a two and a four-qubit non-maximally entangled state where the parties intending to remotely prepare the respective states are not initially entangled. There is a controller of the protocol who oversees the performances of other parties and acts to signal for the execution of the final step in the protocol. There is a Mentor whose action creates entanglement between the rest of the parties and also determines one of the several possible courses of the communication scheme. After that the Mentor quits. The effect of three different noises, namely, Bit-flip, Phase-flip and Amplitude-damping noises are analyzed using the Kraus operator on the otherwise perfect protocol. The decreased fidelity in the presence of noise is numerically studied with respect to noise and other parameters. It is found that in all the three cases the fidelity tends to one as the noise parameter tends to zero.On the representations of Bell's operators in quantum mechanicshttps://zbmath.org/1528.810482024-03-13T18:33:02.981707Z"Sorella, S. P."https://zbmath.org/authors/?q=ai:sorella.silvio-paoloSummary: We point out that, when the dimension of the Hilbert space is greater than two, Bell's operators entering the Bell-CHSH inequality do exhibit inequivalent unitary matrix representations. Although the Bell-CHSH inequality turns out to be violated, the size of the violation is different for different representations, the maximum violation being given by Tsirelson's bound. The feature relies on a pairing mechanism between the modes of the Hilbert space of the system.Quantum sheaf cohomology on Grassmannianshttps://zbmath.org/1528.811582024-03-13T18:33:02.981707Z"Guo, Jirui"https://zbmath.org/authors/?q=ai:guo.jirui"Lu, Zhentao"https://zbmath.org/authors/?q=ai:lu.zhentao"Sharpe, Eric"https://zbmath.org/authors/?q=ai:sharpe.eric-rIn this article under reviewed, the authors describes and investigates quantum sheaf cohomolgy on Grassmanians with deformations of the tangent bundle. A ring structure is used to derive the main results. Quantum cohomology is an essential part in algebraic geometry and string theory. In particular, quantum sheaf cohomology is a generalization quantum cohomology. Many authors studied in details quantum sheaf cohomolgy on toric varieties. There are many stages to tackle and to deal with quantum sheaf cohomolgy. The first author of this article under reviewed \textit{J. Guo} has published an interesting and similar article in the title: [Commun. Math. Phys. 374, No. 2, 661--688 (2020; Zbl 1435.81130)]
A general description of the ring structure has been used and found from both physical prospective and mathematical prospectives. In fact, as was indicates in the article mentioned above, that one can study quantum sheaf cohomology by representing the theory with generators and relations.
The article is well written. It gives crucial and vital methods to focus on physics derivations with excellent examples. These examples describe the correlation functions and quantum cohomollogy in certain specific situation. For the purpose to explain such examples, the authors consider only some special cases.
The article contains interesting sections. Without going in the technical details, this article under reviewed introduces a novel study and a significant approach to quantum sheaf cohomology on Grassmannians. In this article, the introduction provides sufficient background. Such introduction includes the relevant references. There are very good references in the end of the article. The methods of the article are clearly and adequately described. The research design is excellent. In fact, theory of Grassmannians is presented clearly. Non-abelain case is recalled in a good way. Then an interesting section on ring structures of quantum sheaf cohomolgy is presented clearly. This section of ring structure approach contains three subsection. The first is about Gauge invariant operators. Second one is about quantum sheaf cohomology ring with specialization to ordinary classical cohomolgy as well as specialization to quantum cohomology. Then a very excellent section for examples. The section of conclusion is presented. It support the results of the paper. In fact, this theory can be investigated more in the future either by physical approach or mathematical approach.
Reviewer: Ahmad Alghamdi (Makkah)The position-momentum commutator as a generalized function: resolution of the apparent discrepancy between continuous and discrete baseshttps://zbmath.org/1528.811702024-03-13T18:33:02.981707Z"Boykin, Timothy B."https://zbmath.org/authors/?q=ai:boykin.timothy-bSummary: It has been known for many years that the matrix representation of the one-dimensional position-momentum commutator calculated with the position and momentum matrices in a finite basis is not proportional to the diagonal matrix, contrary to what one expects from the continuous-space commutator. This discrepancy has correctly been ascribed to the incompleteness of any finite basis, but without the details of exactly why this happens. Understanding why the discrepancy occurs requires calculating the position, momentum, and commutator matrix elements in the continuous position basis, in which all are generalized functions. The reason for the discrepancy is revealed by replacing the generalized functions with sequences approaching them as their parameter approaches zero. Besides explaining the discrepancy in the discrete and continuous models, this investigation finds an unusual double-peaked sequence for the Dirac delta function.Integral and differential structures for quantum field theoryhttps://zbmath.org/1528.811852024-03-13T18:33:02.981707Z"Labuschagne, Louis"https://zbmath.org/authors/?q=ai:labuschagne.louis-e"Majewski, Adam"https://zbmath.org/authors/?q=ai:majewski.adam-w|majewski.adam-aSummary: The aim of this work is to firstly demonstrate the efficacy of the recently proposed Orlicz space formalism for Quantum theory [\textit{W. A. Majewski} and \textit{L. E. Labuschagne}, Ann. Henri Poincaré 15, No. 6, 1197--1221 (2014; Zbl 1295.82019)], and secondly to show how noncommutative differential structures may naturally be incorporated into this framework. To start off with we specifically propose regularity conditions which in the context of local algebras corresponding to Minkowski space, ensure good behaviour of field operators as observables, and then show that fields obtained by the Osterwalder-Schrader reconstruction theorem are regular in this sense. This complements earlier work by \textit{D. Buchholz} [J. Math. Phys. 31, No. 8, 1839--1846 (1990; Zbl 0708.46061)], \textit{W. Driessler} et al. [Commun. Math. Phys. 105, 49--84 (1986; Zbl 0595.46062)], etc, on generalized \(H\)-bounds. The pair of Orlicz spaces we explicitly use for this purpose, are respectively built on the exponential function (for the description of regular field operators) and on an entropic type function (for the description of the corresponding states). This formalism has been shown to be well suited to a description of quantum statistical mechanics, and in the present work we show that it is also a very useful and elegant tool for Quantum Field Theory. We then introduce the class of tangentially conditioned algebras, which is a large class of local algebras corresponding to globally hyperbolic Lorentzian manifolds that locally ``look like'' the local algebras of Minkowski space. On the one hand this ensures that at a local level, the Orlicz space formalism discussed above is also relevant for a much more general class of local algebras. On the other hand, the structure of this class of algebras, allows for the development of a non-commutative differential geometric structure along the lines of the du Bois-Violette approach to such a theory. In this way we obtain a complete depiction: integrable structures based on local algebras provide a static setting for an analysis of Quantum Field Theory and an effective tool for describing regular behaviour of field operators, whereas differentiable structures posit indispensable tools for a description of equations of motion.A functional approach to the Van der Waals interactionhttps://zbmath.org/1528.812282024-03-13T18:33:02.981707Z"Fosco, C. D."https://zbmath.org/authors/?q=ai:fosco.cesar-daniel"Hansen, G."https://zbmath.org/authors/?q=ai:hansen.gerdSummary: We evaluate the quantum interaction energy between two neutral atoms, using a functional integral approach. In our setup, each atom has an electron bound to the nucleus via a harmonic potential. The resulting expression for the vacuum energy becomes the Van der Waals interaction at the first non-trivial order in an expansion in powers of the fine structure constant, encompassing both the long and short distance behaviours. We also explore the opposite, strong-coupling limit, which yields a result for the interaction energy as well as a threshold for the existence of a vacuum decay probability, manifested here as an imaginary part for the effective action. In the weak-coupling limit, we also study the effect of using a general central potential for the internal structure of the atoms.From symmetries to commutant algebras in standard Hamiltonianshttps://zbmath.org/1528.812292024-03-13T18:33:02.981707Z"Moudgalya, Sanjay"https://zbmath.org/authors/?q=ai:moudgalya.sanjay"Motrunich, Olexei I."https://zbmath.org/authors/?q=ai:motrunich.olexei-iSummary: In this work, we revisit several families of standard Hamiltonians that appear in the literature and discuss their symmetries and conserved quantities in the language of commutant algebras. In particular, we start with families of Hamiltonians defined by parts that are local, and study the algebra of operators that separately commute with each part. The families of models we discuss include the spin-1/2 Heisenberg model and its deformations, several types of spinless and spinful free-fermion models, and the Hubbard model. This language enables a decomposition of the Hilbert space into dynamically disconnected sectors that reduce to the conventional quantum number sectors for regular symmetries. In addition, we find examples of non-standard conserved quantities even in some simple cases, which demonstrates the need to enlarge the usual definitions of symmetries and conserved quantities. In the case of free-fermion models, this decomposition is related to the decompositions of Hilbert space via irreducible representations of certain Lie groups proposed in earlier works, while the algebra perspective applies more broadly, in particular also to arbitrary interacting models. Further, the von Neumann Double Commutant Theorem (DCT) enables a systematic construction of local operators with a given symmetry or commutant algebra, potentially eliminating the need for ``brute-force'' numerical searches carried out in the literature, and we show examples of such applications of the DCT. This paper paves the way for both systematic construction of families of models with exact scars and characterization of such families in terms of non-standard symmetries, pursued in a parallel paper \textit{S. Moudgalya} and \textit{O. I. Motrunich} [``Exhaustive characterization of quantum many-body scars using commutant algebras'', Preprint, \url{arXiv:2209.03377}].Formalism for stochastic perturbations and analysis in relativistic starshttps://zbmath.org/1528.830032024-03-13T18:33:02.981707Z"Satin, Seema"https://zbmath.org/authors/?q=ai:satin.seema-eSummary: Perturbed Einstein's equations with a linear response relation and a stochastic source, applicable to a relativistic star model are worked out. These perturbations which are stochastic in nature, are of significance for building a non-equilibrium statistical mechanics theory in connections with relativistic astrophysics. A fluctuation dissipation relation for a spherically symmetric star in its simplest form is obtained. The FD relation shows how the random velocity fluctuations in the background of the unperturbed star can dissipate into Lagrangian displacement of fluid trajectories of the dense matter. Interestingly in a simple way, a constant (in time) coefficient of dissipation is obtained without a delta correlated noise. This formalism is also extended for perturbed TOV equations which have a stochastic contribution, and show up in terms of the effective or root mean square pressure perturbations. Such contributions can shed light on new ways of analysing the equation of state for dense matter. One may obtain contributions of first and second order in the equation of state using this stochastic approach.Polarized gravitational waves in the parity violating scalar-nonmetricity theoryhttps://zbmath.org/1528.830112024-03-13T18:33:02.981707Z"Chen, Zheng"https://zbmath.org/authors/?q=ai:chen.zheng|chen.zheng.1"Yu, Yang"https://zbmath.org/authors/?q=ai:yu.yang"Gao, Xian"https://zbmath.org/authors/?q=ai:gao.xian(no abstract)Conversion of gravitational and electromagnetic waves without any external background fieldhttps://zbmath.org/1528.830152024-03-13T18:33:02.981707Z"Mishima, Takashi"https://zbmath.org/authors/?q=ai:mishima.takashi"Tomizawa, Shinya"https://zbmath.org/authors/?q=ai:tomizawa.shinyaSummary: Applying a simple harmonic map method to the cylindrically symmetric Einstein-Maxwell system, we obtain exact solutions representing strong nonlinear interaction between gravitational waves and electromagnetic waves in the case without any background field. As an interesting fact, we can show that with adjusted parameters the solution represents occurrences of large conversion phenomena in the intense region of fields near the cylindrically symmetric axis.A note on the canonical formalism for gravityhttps://zbmath.org/1528.830412024-03-13T18:33:02.981707Z"Witten, Edward"https://zbmath.org/authors/?q=ai:witten.edwardSummary: We describe a simple gauge-fixing that leads to a construction of a quantum Hilbert space for quantum gravity in an asymptotically Anti de Sitter spacetime, valid to all orders of perturbation theory. The construction is motivated by a relationship of the phase space of gravity in asymptotically Anti de Sitter spacetime to a cotangent bundle. We describe what is known about this relationship and some extensions that might plausibly be true. A key fact is that, under certain conditions, the Einstein Hamiltonian constraint equation can be viewed as a way to gauge fix the group of conformal rescalings of the metric of a Cauchy hypersurface. An analog of the procedure that we follow for Anti de Sitter gravity leads to standard results for a Klein-Gordon particle.Ladder symmetries of black holes and de Sitter space: Love numbers and quasinormal modeshttps://zbmath.org/1528.830692024-03-13T18:33:02.981707Z"Berens, Roman"https://zbmath.org/authors/?q=ai:berens.roman"Hui, Lam"https://zbmath.org/authors/?q=ai:hui.lam"Sun, Zimo"https://zbmath.org/authors/?q=ai:sun.zimo(no abstract)Impact of multiple modes on the evolution of self-interacting axion condensate around rotating black holeshttps://zbmath.org/1528.830812024-03-13T18:33:02.981707Z"Omiya, Hidetoshi"https://zbmath.org/authors/?q=ai:omiya.hidetoshi"Takahashi, Takuya"https://zbmath.org/authors/?q=ai:takahashi.takuya"Tanaka, Takahiro"https://zbmath.org/authors/?q=ai:tanaka.takahiro"Yoshino, Hirotaka"https://zbmath.org/authors/?q=ai:yoshino.hirotaka(no abstract)Hidden freedom in the mode expansion on static spacetimeshttps://zbmath.org/1528.831342024-03-13T18:33:02.981707Z"de Souza Campos, Lissa"https://zbmath.org/authors/?q=ai:de-souza-campos.lissa"Dappiaggi, Claudio"https://zbmath.org/authors/?q=ai:dappiaggi.claudio"Sinibaldi, Luca"https://zbmath.org/authors/?q=ai:sinibaldi.lucaSummary: We review the construction of ground states focusing on a real scalar field whose dynamics is ruled by the Klein-Gordon equation on a large class of static spacetimes. As in the analysis of the classical equations of motion, when enough isometries are present, via a mode expansion the construction of two-point correlation functions boils down to solving a second order, ordinary differential equation on an interval of the real line. Using the language of Sturm-Liouville theory, most compelling is the scenario when one endpoint of such interval is classified as a limit circle, as it often happens when one is working on globally hyperbolic spacetimes with a timelike boundary. In this case, beyond initial data, one needs to specify a boundary condition both to have a well-defined classical dynamics and to select a corresponding ground state. Here, we take into account boundary conditions of Robin type by using well-known results from Sturm-Liouville theory, but we go beyond the existing literature by exploring an unnoticed freedom that emerges from the intrinsic arbitrariness of secondary solutions at a limit circle endpoint. Accordingly, we show that infinitely many one-parameter families of sensible dynamics are admissible. In other words, we emphasize that physical constraints guaranteeing the construction of ground states do not, in general, fix one such state unambiguously. In addition, we provide, in full detail, an example on \((1+1)\)-half Minkowski spacetime to spell out the rationale in a specific scenario where analytic formulae can be obtained.Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivativeshttps://zbmath.org/1528.901902024-03-13T18:33:02.981707Z"Khanh, Pham Duy"https://zbmath.org/authors/?q=ai:khanh.pham-duy"Phat, Vo Thanh"https://zbmath.org/authors/?q=ai:phat.vo-thanhThis compact paper is concerned with establishing second-order necessary and sufficient conditions for the (strict) quasiconvexity and the (strict) pseudoconvexity of \(\mathcal{C}^{1, 1}\)-smooth functions on finite-dimensional Euclidean spaces. These second-order conditions are also effective and important to various real-world situations. The paper is well-organized including the discussions on the necessary and sufficient conditions in Sections 3 and 4 correspondingly. Both Fréchet and Mordukhovich's second-order subdifferentials are discussed. The authors also present many examples in Section 4 to analyze and illustrate their results.
Reviewer: Hongpeng Sun (Beijing)Projections onto hyperbolas or bilinear constraint sets in Hilbert spaceshttps://zbmath.org/1528.901952024-03-13T18:33:02.981707Z"Bauschke, Heinz H."https://zbmath.org/authors/?q=ai:bauschke.heinz-h"Krishan Lal, Manish"https://zbmath.org/authors/?q=ai:krishan-lal.manish"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfuSummary: Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space.Essential stability in unified vector optimizationhttps://zbmath.org/1528.902452024-03-13T18:33:02.981707Z"Kapoor, Shiva"https://zbmath.org/authors/?q=ai:kapoor.shiva"Lalitha, C. S."https://zbmath.org/authors/?q=ai:lalitha.c-sThe paper focuses on several properties of the essential efficient solutions, the essential sets and the essential components of a vector optimization problem. These concepts are based on perturbations of the objective function and the feasible set of the nominal problem. Namely, the framework of the involved problems considers objective functions from a metric space to a normed space \(Y\) and compact feasible sets. In addition, an arbitrary nonempty proper domination set \(D\subset Y\) is assumed, although in the main results some additional properties are supposed, for instance, \(D\) closed, \(D+D\subseteq D\) and other ones.
The main contributions of the paper concern the characterization of an efficient solution to be essential and a sufficient condition that implies the efficient set to be an essential set, through the lower and upper semicontinuity of the efficient solution mapping, respectively; the density of the set of stable vector optimization problems in the sense of Baire category; sufficient conditions for the existence of essential sets, minimal essential sets and essential components; finally, characterizations of an arbitrary nonempty closed subset of efficient solutions to be an essential set, and a component of efficient solutions to be an essential one.
Reviewer: César Gutiérrez (Valladolid)Second order optimality conditions for minimization on a general set. I: Applications to mathematical programminghttps://zbmath.org/1528.902752024-03-13T18:33:02.981707Z"Frankowska, Hélène"https://zbmath.org/authors/?q=ai:frankowska.helene"Osmolovskii, Nikolai P."https://zbmath.org/authors/?q=ai:osmolovskii.nikolai-pavlovichSummary: This paper is devoted to second-order optimality conditions for minimization of a \(C^2\) function \(f\) on a general set \(K\) in a Banach space \(X\). We consider both necessary and sufficient conditions of the second-order which differ by the strengthening of inequalities in their formulations. The conditions use first and second order approximations (first and second-order tangents) of the set \(K\). The no gap sufficient conditions need additional assumptions in comparison with necessary conditions. We show that these assumptions hold true in the case when the set \(K\) is an intersection of a finite number of sets described by smooth inequalities and equalities, like in problems of the mathematical programming. Moreover, we illustrate the new conditions by deducing some mathematical programming results. In this sense the paper is partly a survey. One non-trivial illustrative example in an infinite dimensional space concerns the case when \(K\) can not be represented as an intersection described above. The novelty of our approach is due, on one hand, to the arbitrariness of the set \(K\), and on the other hand, to quite straightforward proofs.Random average sampling in a reproducing kernel subspace of mixed Lebesgue space \(L^{p,q}(\mathbb{R}^{n+1})\)https://zbmath.org/1528.940232024-03-13T18:33:02.981707Z"Patel, Dhiraj"https://zbmath.org/authors/?q=ai:patel.dhiraj"Sivananthan, S."https://zbmath.org/authors/?q=ai:sivananthan.sivalingamSummary: In this paper, we study average sampling inequality in a probabilistic framework for a reproducing kernel subspace \(V\) of mixed Lebesgue space. More precisely, we show with high probability that a function concentrated on a compact cube \(C\) can be stably recovered from their \(\mathcal{O}(\mu (C)\log \mu(C))\) many average values at uniformly distributed random points over \(C\), where \(\mu\) is a Lebesgue measure. Further, we propose an exponential convergence reconstruction scheme to reconstruct the concentrated function from their random average measurements and illustrate with an example.