Recent zbMATH articles in MSC 46Bhttps://zbmath.org/atom/cc/46B2022-07-25T18:03:43.254055ZWerkzeugEstimates on the Markov convexity of Carnot groups and quantitative nonembeddabilityhttps://zbmath.org/1487.220082022-07-25T18:03:43.254055Z"Gartland, Chris"https://zbmath.org/authors/?q=ai:gartland.chrisSummary: We show that every graded nilpotent Lie group \(G\) of step \(r\), equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot groups with Carnot-Caratheodory metric) is Markov \(p\)-convex for all \(p \in [2r, \infty)\). We also show that this is sharp whenever \(G\) is a Carnot group with \(r \leq 3\), a free Carnot group, or a jet space group; such groups are not Markov \(p\)-convex for any \(p \in(0, 2 r)\). This continues a line of research started by Li who proved this sharp result when \(G\) is the Heisenberg group. As corollaries, we obtain new estimates on the non-biLipschitz embeddability of some finitely generated nilpotent groups into nilpotent Lie groups of lower step. Sharp estimates of this type are known when the domain is the Heisenberg group and the target is a uniformly convex Banach space or \(L^1\), but not when the target is a nonabelian nilpotent group.Existence of solutions for infinite systems of regular fractional Sturm-Liouville problems in the spaces of tempered sequenceshttps://zbmath.org/1487.340352022-07-25T18:03:43.254055Z"Prasad, K. Rajendra"https://zbmath.org/authors/?q=ai:prasad.kapula-rajendra"Khuddush, Mahammad"https://zbmath.org/authors/?q=ai:khuddush.mahammad"Leela, D."https://zbmath.org/authors/?q=ai:leela.dSummary: In this paper we investigate the existence of solutions for infinite systems of regular fractional Sturm-Liouville problems by an application of Meir-Keeler fixed point theorem in the tempered sequence spaces. We provide examples to check validity of our obtained results.Divergence a.e. of the greedy algorithm by Faber-Schauder system for continuous functionshttps://zbmath.org/1487.410282022-07-25T18:03:43.254055Z"Grigoryan, M. G."https://zbmath.org/authors/?q=ai:grigoryan.martin-g"Sargsyan, A. A."https://zbmath.org/authors/?q=ai:sargsyan.artsrun-a(no abstract)Properties of suns in the spaces \(L^1\) and \(C(Q)\)https://zbmath.org/1487.410422022-07-25T18:03:43.254055Z"Tsar'kov, I. G."https://zbmath.org/authors/?q=ai:tsarkov.igor-gLet \(X\) be a normed linear space with dual \(X^*\) and let \(B\, (S), B^*\, (S^*)\) be the unit balls (unit spheres) of the spaces \(X\) and \(X^*\), respectively. A monotone path is a continuous mapping \(k:[0,1]\to X\) such that \(\varphi\circ k\) is a monotone function for every \(\varphi \in\mathrm{ext}\, S^*\) (the set of extreme points of \(S^*\)). A subset \(M\) of a normed space \(X\) is called linearly monotone connected if any two points in \(M\) can be joined by a monotone path contained in \(M\). This is a notion lying between linear connectedness and convexity. A point \(x\in X\setminus M\) is called a solar (strictly solar) point of \(M\) if \(P_M(x)\ne\emptyset\) and \(y\in P_M((1-\lambda)x+\lambda y)\) for some \(y\in P_M(x)\) (resp. for all \(y\in P_M(x)\)), where \(P_M\) denotes the (set-valued) metric projection on \(M\). The set \(M\) is called a sun (strict sun) if every point in \(X\setminus M\) is a solar (strict solar) point for \(M\).
For \(x,y\in M\) let \(\mathbf{m}(x,y)\) be the intersection of all closed balls containing the points \(x,y\) and let \[[\![x,y]\!]=\{z\in X:\min\{\varphi(x),\varphi(y)\}\le\varphi(z)\le\max\{\varphi(x),\varphi(y)\},\;\forall \varphi\in\mathrm{ext}\, S^*\}\,.\] The set \(M\) is called Menger connected (strongly Menger connected) if \(M\,\cap\, \mathbf{m}(x,y)\ne\{x,y\}\) (resp. \( M\,\cap \,[\![x,y]\!]\ne\{x,y\}\)), for all distinct points \(x,y\in M\). Since \([\![x,y]\!]\subset \mathbf{m}(x,y)\), the strong Menger connectedness implies ordinary Menger connectedness. If the space \(X\) is separable, then these notions coincide. It is unknown whether this holds or not in non-separable normed spaces.
These notions are involved in the study of convexity of Chebyshev set, see [\textit{A. R. Alimov} and \textit{I. G. Tsar'kov}, Russ. Math. Surv. 71, No. 1, 1--77 (2016; Zbl 1350.41031); translation from Usp. Mat. Nauk 71, No. 1, 3--84 (2016)].
The present paper is concerned with suns in the classical spaces \(L^1\) and \(C(Q)\).
One shows that every boundedly compact sun in \(L^1[0,1]\) is convex and, as the author remarks, the result holds in the space \(L^1(\mu)\) for any non-atomic measure \(\mu\).
Recall that a set \(K\) is called boundedly compact if its intersection with every closed ball is compact. If \(K\) is a sun in \(L^1[0,1]\) such that there exists a closed ball \(B(x_0,r_0)\) with \(x_0\in K\) such that \(K\cap B(x_0,r_0)\) is compact, then \(K\) is a boundedly compact convex set (Theorem 2).
In the space \(C(Q)\), where \(Q\) is a Hausdorff compact topological space, every boundedly weakly compact sun is strongly Menger connected, and so monotone path connected (Theorem 4).
The assertion concerning the monotone path connectedness follows by a result of [\textit{A. R. Alimov}, Izv. Math. 78, No. 4, 641--655 (2014; Zbl 1303.41018); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 4, (2014)]: every boundedly weakly compact Menger connected subset of \(C(Q)\) is monotone path connected.
Reviewer: Stefan Cobzaş (Cluj-Napoca)Interpolation theory for the HK-Fourier transformhttps://zbmath.org/1487.420082022-07-25T18:03:43.254055Z"Arredondo, Juan H."https://zbmath.org/authors/?q=ai:arredondo.juan-h"Reyes, Alfredo"https://zbmath.org/authors/?q=ai:reyes.alfredoAuthors' abstract: We use the Henstock-Kurzweil integral and interpolation theory to extend the Fourier cosine transform operator, broadening some classical properties such as the Riemann-Lebesgue lemma. Furthermore, we show that a qualitative difference between the cosine and sine transform is preserved on differentiable functions.
Reviewer: Vishvesh Kumar (Delhi)On some operators and dilations of frame generator and dual pair of frame generators of two structured unitary systemshttps://zbmath.org/1487.420742022-07-25T18:03:43.254055Z"Guo, Xunxiang"https://zbmath.org/authors/?q=ai:guo.xunxiang"Chen, Yonghong"https://zbmath.org/authors/?q=ai:chen.yonghongThis paper is on the study of dilations of frame generators and parts of dual frame generators in the settings of product unitary systems and group-like unitary systems. A product unitary system is of the form \(\mathcal{U} = \mathcal{U}_1\mathcal{U}_0\) with \(\mathcal{U}_1\) and \(\mathcal{U_0}\) being unitary groups of operators on a Hilbert space \(\mathcal{H}\) such that \(\mathcal{U}_1\cap \mathcal{U}_0 = \{I\}\), while a group-like unitary system \(\mathcal{U}\) is such that \(\text{group}(\mathcal{U}) \subset \mathbb{T}\mathcal{U}:=\{\lambda U: \lambda\in \mathbb{T}, U\in \mathcal{U}\}\) and \(\mathcal{U}\) is linearly independent in the sense that \(\mathbb{T}U\neq \mathbb{T}V\) whenever \(U\neq V\) for \(U,V\in \mathcal{U}\). They are the abstract models of wavelet frames and Gabor frames, respectively. Here \(\mathbb{T}\) is the unit circle.
For the product unitary system, the paper proves two main results. One is that if \(\mathcal{U}_0\eta:=\{U_0\eta: U_0\in\mathcal{U}_0\} \) is a frame for \([\mathcal{U}_0\eta]:=\overline{\mathrm{span}}\{\mathcal{U_0}\eta\}\), \([\mathcal{U}_0\eta]\) is a complete wandering subspace for \(\mathcal{U_1}\), and \(U\) is a unitary operator from the \(C^\ast\)-algebra \(W^\ast(\mathcal{U}_0)\) generated by \(\mathcal{U}_0\), then \(U\eta\) is a frame generator of \(\mathcal{U}\), that is, \(\{V(U\eta): V\in\mathcal{U}\}\) is a frame for the underlying Hilbert space \(\mathcal{H}\). The other one is that if there exists an \(x\in \mathcal{H}\) such that \(\mathcal{U}x\) is an orthonormal basis for \(\mathcal{H}\), \(\mathcal{U}_0\eta\) is a normalized tight frame for \([\mathcal{U}_0\eta]\), and \([\mathcal{U}_0\eta]\) is a complete wandering space subspace for \(\mathcal{U}_1\), then there exists a \(\xi\in\mathcal{H}\) such that \(\mathcal{U}\eta\oplus\mathcal{U}\xi\) is an orthonormal basis for \(\mathcal{H}\oplus [\mathcal{U}\xi]\).
For the group-like unitary system, the paper proves several results including (1) if \(\mathcal{U}\eta\) is a frame and \(S_\eta\) is the frame operator of \(\mathcal{U}\eta\), i.e., \(S_\eta(x)=\sum_{U\in\mathcal{U}}\langle x, U\eta\rangle U\eta\), then \(S_\eta\) commutes with all \(V\in \mathcal{U}\), (2) if \(\mathcal{U}\psi\) is an orthonormal basis and \(\mathcal{U}\eta\) is a frame for \(\mathcal{H}\), and \(T\) is given by \(T(x) :=\sum_{U\in\mathcal{U}}\langle x, U\eta\rangle U\psi\), then \(T\), \(T^\ast\) both commute with all \(V\in \mathcal{U}\), (3) If \(\mathcal{U}\eta\) is a frame, then its cannonical dual has the same frame structure, (4) if there exists an \(x\in \mathcal{H}\) such that \(\mathcal{U}x\) is an orthonormal basis for \(\mathcal{H}\), \(\mathcal{U}\eta\) is a normalized tight frame for \(\mathcal{H}\), then there exists a \(\xi\in\mathcal{H}\) such that \(\mathcal{U}\eta\oplus\mathcal{U}\xi\) is an orthonormal basis for \(\mathcal{H}\oplus [\mathcal{U}\xi]\), (5) if one relexes the condition that \(\mathcal{U}\eta\) is a normalized tight frame for \(\mathcal{H}\) in (4) to \(\mathcal{U}\eta\) is a frame for \(\mathcal{H}\), then there exists a \(\xi\in\mathcal{H}\) such that \(\mathcal{U}\eta\oplus\mathcal{U}\xi\) is an Riesz basis for \(\mathcal{H}\oplus [\mathcal{U}\xi]\), (6) if there exists an \(x\in \mathcal{H}\) such that \(\mathcal{U}x\) is an orthonormal basis for \(\mathcal{H}\) and \((\mathcal{U}\varphi,\mathcal{U}\eta)\) is a pair of dual frames for \(\mathcal{H}\), then there exist a larger Hilbert space \(\mathcal{K}\supset \mathcal{H}\) and a pair of dual Riesz bases \((\mathcal{U}\Phi, \mathcal{U}\tilde\Phi)\) such that the \(P_{\mathcal{H}}(U\Phi) = U\phi\) and \(P_{\mathcal{H}}(U\tilde \Phi) = U\eta\) for all \(U\in \mathcal{U}\).
Reviewer: Xiaosheng Zhuang (Hong Kong)A note on Nakano generalized difference sequence spacehttps://zbmath.org/1487.460032022-07-25T18:03:43.254055Z"Bakery, Awad A."https://zbmath.org/authors/?q=ai:bakery.awad-a"Elmatty, Afaf R. Abou"https://zbmath.org/authors/?q=ai:abou-elmatty.afaf-r(no abstract)Sequence spaces derived by the triple band generalized Fibonacci difference operatorhttps://zbmath.org/1487.460052022-07-25T18:03:43.254055Z"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-abdul|mohiuddine.syed-adbul"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamal(no abstract)Almost minimal orthogonal projectionshttps://zbmath.org/1487.460072022-07-25T18:03:43.254055Z"Basso, Giuliano"https://zbmath.org/authors/?q=ai:basso.giulianoThis paper makes a significant contribution to the problem of finding the maximum projection constant for the class of Banach spaces of a given finite dimension. More precisely, for any integer \(n\) and \(\varepsilon>0\), it shows that there is an integer \(d>n\) and an \(n\)-dimensional subspace of \(\mathbb{R}^d\) (equipped with the \(\ell_1\) norm) whose relative projection constant is almost maximal (i.e., within \(\varepsilon\) of the maximum). Moreover, the orthogonal projection (with respect to a suitable orthonormal basis of \(\mathbb{R}^d\)) onto this subspace has almost minimal norm.
Reviewer: David Yost (Ballarat)Isometric embeddability of \(S_q^m\) into \(S_p^n\)https://zbmath.org/1487.460082022-07-25T18:03:43.254055Z"Chattopadhyay, Arup"https://zbmath.org/authors/?q=ai:chattopadhyay.arup"Hong, Guixiang"https://zbmath.org/authors/?q=ai:hong.guixiang"Pal, Avijit"https://zbmath.org/authors/?q=ai:pal.avijit"Pradhan, Chandan"https://zbmath.org/authors/?q=ai:pradhan.chandan"Ray, Samya Kumar"https://zbmath.org/authors/?q=ai:ray.samya-kumarMotivated by the classical research on linear isometric embeddability of \(\ell_q^m\) into \(\ell_p^n\) [\textit{Yu. I. Lyubich} and \textit{O. A. Shatalova}, St. Petersbg. Math. J. 16, No. 1, 9--24 (2004; Zbl 1076.46005)], the authors undertake a detailed study of similar questions for finite dimensional Schatten \(p\)-classes. Using a spectrum of ideas ranging from the Kato-Rellich theorem and using other directions, like the Birkhoff-James orthogonality, several results enunciating the importance of the summability index of the domain space being \(2,3\), are obtained by a delicate case-by-case analysis.
We only quote a couple of results from this long and interesting paper. If there is a linear isometry of \(S_q^m\) into \(S_1^n\) (for \(1<q< \infty\), \(2 \leq m \leq n\)), then \(q = 2\) or \(q=3\). Similarly, when the range is \(S_p^n\) (\(n < \infty\)) and \((q,p) \in [1, \infty]\times [1,\infty)\), then \(q=2\).
Reviewer: T.S.S.R.K. Rao (Bangalore)A spherical version of the Kowalski-Słodkowski theorem and its applicationshttps://zbmath.org/1487.460092022-07-25T18:03:43.254055Z"Oi, Shiho"https://zbmath.org/authors/?q=ai:oi.shihoSummary: \textit{L.~Li} et al. [Publ. Mat., Barc. 63, No.~1, 241--264 (2019; Zbl 1419.46011]
generalized the Kowalski-Słodkowski theorem by establishing the following spherical variant: let \(A\) be a unital complex Banach algebra and let \(\Delta : A \to \mathbb{C}\) be a mapping satisfying the following properties:
\begin{itemize}
\item[(a)] \( \Delta\) is 1-homogeneous (that is, \( \Delta (\lambda x)=\lambda \Delta (x)\) for all \(x \in A\), \(\lambda \in \mathbb C )\);
\item[(b)] \(\Delta (x)-\Delta (y) \in \mathbb{T}\sigma (x-y)\), \(x,y \in A\).
\end{itemize}
Then \(\Delta\) is linear and there exists \(\lambda_0 \in \mathbb{T}\) such that \(\lambda_0\Delta\) is multiplicative. In this note we prove that if (a) is relaxed to \(\Delta (0)=0\), then \(\Delta\) is complex-linear or conjugate-linear and \(\overline{\Delta (\boldsymbol{1})}\Delta\) is multiplicative. We extend the Kowalski-Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by
\textit{L.~Molnár} [J. Math. Anal. Appl. 479, No.~1, 569--580 (2019; Zbl 07096850)]
and also in a private communication between \textit{L.~Molnár} and \textit{O.~Hatori}, 2018.Stability of symmetric \(\varepsilon \)-isometries on wedgeshttps://zbmath.org/1487.460102022-07-25T18:03:43.254055Z"Sun, Longfa"https://zbmath.org/authors/?q=ai:sun.longfaSummary: Let \(X, Y\) be Banach spaces, \(W\) be a closed wedge of \(X\), and \(f:W\cup -W\rightarrow Y\) be a symmetric \(\varepsilon \)-isometry. We firstly establish a weak stability formula about the symmetric isometry \(f\). Making use of it, we prove a series of new stability theorems for the symmetric isometries defined on the positive cones \(W\) of \(C(K)\)-spaces. For example, if \(f(W\cup -W)\) contains a reproducing wedge of \(Y\), then there exists a linear surjective isometry \(U:C(K)\rightarrow Y\) such that \(f-U\) is uniformly bounded by \(\frac{3}{2}\varepsilon\) on \(W\cup -W\); and if \(\overline{\text{co}}f(W\cup -W)\) contains a reproducing wedge \(P\) of \(Y\), then there exists a bounded linear operator \(T:Y\rightarrow C(K)\) with \(\Vert T\Vert =1\) such that
\[ \Vert Tf(x)-x\Vert \le \frac{3}{2}\varepsilon,\text{ for all } x\in W\cup -W. \]Hyers-Ulam stability of \(\varepsilon \)-isometries between the positive cones of \(c_0\)https://zbmath.org/1487.460112022-07-25T18:03:43.254055Z"Sun, Longfa"https://zbmath.org/authors/?q=ai:sun.longfaLet $X$ be a Banach space over the reals. A wedge $W$ in $X$ is a convex set which is closed under multiplication by non-negative scalars. The paper under review investigates the following problem:
Let $X, Y$ be two Banach spaces, $W_1\subset X$, $W_2\subset Y$ be two wedges, $f\colon W_1\rightarrow W_2$ be a standard surjective $\varepsilon$-isometry. Do there exist a surjective (additive) isometry $V\colon W_1\rightarrow W_2$ and $\gamma>0$ such that $$ \|f(x)-V(x)\|\leq \gamma\varepsilon \text{ for all }x\in W_1? $$
The main result of the author proves that the above-mentioned problem has an affirmative answer when $X=Y=c_0$ and $W_1=W_2=c^+_0$, where $c^+_0$ is the positive cone of $c_0$. Moreover, the assumption on the surjectivity of $f$ can be relaxed by requiring $f$ to be $\delta$-surjective for some $\delta\geq 0$.
Reviewer: Johann Langemets (Tartu)Absolutely \((q, 1)\)-summing operators acting in \(C(K)\)-spaces and the weighted Orlicz property for Banach spaceshttps://zbmath.org/1487.460122022-07-25T18:03:43.254055Z"Calabuig, J. M."https://zbmath.org/authors/?q=ai:calabuig.jose-m"Sánchez Pérez, E. A."https://zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonsoThe authors present the main result which is a characterization of the class of homogeneous weights for which the coincidence of weak weighted summable and norm summable sequences in a Banach space implies that it is finite dimensional. In order to do this, the notion of weighted \((q,\phi )\)-Orlicz property for a Banach space is introduced. Then Pisier's Theorem can be proved using the same standard separation argument that proves Pietsch's Theorem and Maurey-Rosenthal's Theorem.
Reviewer: Elhadj Dahia (Bou Saâda)Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measureshttps://zbmath.org/1487.460132022-07-25T18:03:43.254055Z"Alonso-Gutiérrez, David"https://zbmath.org/authors/?q=ai:alonso-gutierrez.david"Prochno, Joscha"https://zbmath.org/authors/?q=ai:prochno.joschaSummary: In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector \(X_n\) being in a thin shell of radius \(\sqrt{n}\) times the asymptotic value of \(n^{-1/2}(\mathbb{E}[\|X_n\|_2^2])^{1/2}\) (as \(n\to\infty)\), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter \(t=t_n\) goes down to zero as the dimension \(n\) of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by
\textit{Z.~Kabluchko} and \textit{J.~Prochno} [J. Math. Anal. Appl. 495, No.~1, Article ID 124687, 19~p. (2021; Zbl 1475.46007)].Note on a generalization of the space of derivatives of Lipschitz functionshttps://zbmath.org/1487.460142022-07-25T18:03:43.254055Z"Kermausuor, S."https://zbmath.org/authors/?q=ai:kermausuor.seth"Kwessi, E."https://zbmath.org/authors/?q=ai:kwessi.eddy-a"De Souza, G."https://zbmath.org/authors/?q=ai:de-souza.geraldo-soaresSummary: In this note, we denote by \((Lip^1)'\) the space of derivatives of Lipschitz functions of order 1. We propose a generalization of the space \((Lip^1)'\) on the interval \([0,2\pi]\) for general measures on subsets of \([0,2\pi]\) with respect to the representation of the norm. As a byproduct, we obtain Hölder's type inequalities and duality results between the space \((Lip^1)'\) as well as its generalization, and the special atoms spaces \(B\) and \(B(\mu,1)\), spaces first introduced by De Souza in his PhD thesis. Another byproduct is a relation between the space \((Lip^1)'\) as well as its generalization, and the space \(L_\infty \). As a result we prove that the special atom space is a simple characterization of \(L_1\).Local interpolation and interpolating baseshttps://zbmath.org/1487.460152022-07-25T18:03:43.254055Z"Goncharov, A. P."https://zbmath.org/authors/?q=ai:goncharov.alexander-p(no abstract)On linear operators that preserve \(BJ\)-orthogonality in 2-normed spacehttps://zbmath.org/1487.460162022-07-25T18:03:43.254055Z"Iranmanesh, Mahdi"https://zbmath.org/authors/?q=ai:iranmanesh.mahdi"Sanatee, Ali Ganjbakhsh"https://zbmath.org/authors/?q=ai:sanatee.ali-ganjbakhshSummary: Let \(X\) be a real 2-Banach space. We follow \textit{G. H. Mashadi} et al. [Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 17, 76--83 (2006; Zbl 1164.46317)] in saying that \(x\) is orthogonal to \(y\) if there exists a subspace \(V\) of \(X\) with \(\operatorname{codim}(V) = 1\) such that \(\| x+\lambda y, z \| \geqslant \| x, z \|\) for every \(z\in V\) and \(\lambda \in\mathbb{R}\). In this paper, we prove that every linear self mapping \(T: X \longrightarrow X\) which preserves orthogonality is a 2-isometry multiplied by a constant.On new moduli related to the generalization of the parallelogram lawhttps://zbmath.org/1487.460172022-07-25T18:03:43.254055Z"Liu, Qi"https://zbmath.org/authors/?q=ai:liu.qi"Zhou, Chuanjiang"https://zbmath.org/authors/?q=ai:zhou.chuanjiang"Sarfraz, Muhammad"https://zbmath.org/authors/?q=ai:sarfraz.muhammad"Li, Yongjin"https://zbmath.org/authors/?q=ai:li.yongjinIn the last few decades, several geometric constants have been defined and studied in the literature, which make it easier for us to deal with some problems in Banach spaces, because it can describe the geometric properties of the space quantitatively, and these geometric constants have mathematical beauty, and there are countless relationships between different geometric constants. Special attention is paid to the von Neumann-Jordan constant and the James constant, which are rigorously studied based on their importance.
The extreme point set plays an important role in Banach space geometry because it can largely reflect the geometric properties of a space \(X\), such as the well-known convexity.
\textit{M. S. Moslehian} and \textit{J. M. Rassias} obtained a new equivalent characterization of inner product spaces [Commun. Math. Anal. 8, No.~2, 16--21 (2010; Zbl 1194.46037)].
The authors introduce a new geometric constant \(L'_{ YJ}(\lambda, μ, X)\) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias [loc.\,cit.]. They start with some basic properties of this new coefficient. Next, it is shown that, for a Banach space, \(L'_{ YJ}(\lambda, \mu, X)\) becomes 1 if and only if the norm is induced by an inner product. Moreover, some sufficient conditions which imply normal structure are presented. At the end, some relations between the well-known geometric constants and \(L'_{ YJ}(\lambda, \mu, X)\) through some inequalities are described.
The paper opens many useful avenues for future research in this area.
Reviewer: V. Lokesha (Bangalore)Some geometric characterizations of a fractional Banach sethttps://zbmath.org/1487.460182022-07-25T18:03:43.254055Z"Özger, Faruk"https://zbmath.org/authors/?q=ai:ozger.farukSummary: This paper is devoted to investigate the modular structure of a fractional Banach set of sequences and prove that this set is reflexive and convex and it possesses uniform Opial, \(( \beta)\), \((L)\) and \((H)\) properties. The convexity of the set is investigated by the notion of extreme points. These properties play an important role both in the study of fixed point theory and in the geometric characterizations of the Banach sets of sequences. This study extends the scope of the fractional calculus and it is related with fixed point and approximation theories.Some remarks on orthogonality of bounded linear operatorshttps://zbmath.org/1487.460192022-07-25T18:03:43.254055Z"Ray, Anubhab"https://zbmath.org/authors/?q=ai:ray.anubhab"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallol"Sain, Debmalya"https://zbmath.org/authors/?q=ai:sain.debmalya"Dey, Subhrajit"https://zbmath.org/authors/?q=ai:dey.subhrajitLet \(X,Y\) be real Banach spaces and let \({\mathcal L}(X,Y)\) denote the space of bounded linear operators. The authors and their collaborators have been studying over a decade aspects of Birkhoff-James orthogonality of two operators \(T,A \in {\mathcal L}(X,Y)\), i.e., \(\|T+\lambda A\| \geq \|T\|\) for all scalars \(\lambda\). It is easy to see that if \(T\) attains its norm at a unit vector \(x_0\) and \(\|T(x_0)+\lambda A(x_0)\| \geq \|T(x_0)\|=\|T\|\) for all scalars \(\lambda\), then the operator inequality holds. The validity of the converse question, for all \(A\), is now called the Bhatia-Šemrl property. See the reviewer's article [Numer. Funct. Anal. Optim. 42, No. 10, 1201--1208 (2021; Zbl 07431317)] for an interpretation of this in terms of min-max formulae.
It is known that the converse question is related to the structure of the set of norm attaining vectors \(M_T\) of \(T\). In this paper it is shown that if \(X\) is \(2\)-dimensional Banach space, \(Y\) is not the scalar field and \(M_T\) has more than two components, then the converse implication fails for \(T\). One way this can happen is when there are independent vectors \(x,y \in M_T\) such that both \(\frac{x+y}{\|x+y\|}\),\(\frac{x-y}{\|x-y\|}\) are not in \(M_T\).
Reviewer: T.S.S.R.K. Rao (Bangalore)Vertices of the unit ball of subspaces in \(\mathcal{L}(H)\) and strong unicity of best approximation in \(\mathcal{L}(l_2^2)\)https://zbmath.org/1487.460202022-07-25T18:03:43.254055Z"Wójcik, Paweł"https://zbmath.org/authors/?q=ai:wojcik.pawelSummary: Let \(\mathcal{L}(H)\) be the space of linear operators acting from the finite-dimensional real Hilbert space into itself. The purpose of this paper is to present results concerning the geometric properties of some subspaces of \(\mathcal{L}(H)\). In particular, vertices of the closed unit ball of those subspaces are discussed. The problem of best approximation in the space \(\mathcal{L}(l_2^2)\) is investigated. Moreover, in this paper we show an example of an effective method seeking a unique minimal projection which is strongly unique.The locally \(k\)-uniformly extremely convex and midpoint locally \(k\)-uniformly extremely convex Banach spaceshttps://zbmath.org/1487.460212022-07-25T18:03:43.254055Z"Wulede, Suyalatu"https://zbmath.org/authors/?q=ai:wulede.suyalatuSummary: The locally \(k\)-uniformly extremely convex Banach spaces are introduced. It is shown here that every locally \(k\)-uniformly extremely convex space is locally \((k+1)\)-uniformly extremely convex space, but the converse implication is not true. Some properties of locally \(k\)-uniformly extremely convex spaces are given. Specially, it is proved that if the second dual of Banach space \(X\) is locally 2-uniformly extremely convex, then \(X\) is reflexive. In addition, the notion of midpoint locally \(k\)-uniformly extreme convexity is introduced and its relations to other type convexity are given.Interpolation of spaces of functions of positive smoothness on a domainhttps://zbmath.org/1487.460222022-07-25T18:03:43.254055Z"Besov, O. V."https://zbmath.org/authors/?q=ai:besov.oleg-vSummary: An interpolation theorem for spaces of functions of positive smoothness on a domain with flexible cone condition is established.Topological and algebraic genericity in chains of sequence spaces and function spaceshttps://zbmath.org/1487.460232022-07-25T18:03:43.254055Z"Bernal-González, L."https://zbmath.org/authors/?q=ai:bernal-gonzalez.luis"Nestoridis, V."https://zbmath.org/authors/?q=ai:nestoridis.vassiliIn the present paper, the authors study topological genericity, algebraic genericity and spaceability of \(\ell^p\) spaces and of intersections of them, as factors in a chain \(X_i \subset X_j, X_i \neq X_j\) for \(i < j\). The aforementioned notions are defined via properties of \(X_j \setminus X_i\) with respect to the space \(X_j\). Namely, in the chain context, these notions are relative to a pair of members of the chain. So, one speaks about topological genericity whenever, \(X_j \setminus X_i\) is a \(G_\delta\)-dense subset of the space \(X_j\) or about algebraic genericity in case, except for \(0\), \(X_j \setminus X_i\) contains a vector space, dense in \(X_j\), while spaceability is gained whenever \(X_j \setminus X_i\) contains except for \(0\), an infinite-dimensional vector space, closed in \(X_j\).
Details of some of the proofs that concern the above three properties for \(\ell^p\) spaces, are included in [the second author, ``A project about chains of spaces regarding topological and algebraic genericity and spaceability'', Preprint (2020), \url{arXiv:2005.01023}], see also [Bull. Am. Math. Soc. 51, No.~1, 71--130 (2014; Zbl 1292.46004)] by the first author et al. Further, the authors study topological or algebraic genericity, and spaceability for other chains with certain types of spaces, or for chains of holomorphic functions, as, e.g., Hardy spaces \(H^p\) on the unit disc. One of the main results, concerning the algebraic genericity of the latter spaces, assures that for \(0<p<q<+\infty\), the set \((H^p \setminus H^q)\cup \{0\}\) contains a vector space which is dense in \(H^p\). The presentation of the results is clear and it is enriched by detailed comments on the form of the used spaces in the referred chains.
Reviewer: Marina Haralampidou (Athína)Multiplier algebras of normed spaces of continuous functionshttps://zbmath.org/1487.460242022-07-25T18:03:43.254055Z"Bilokopytov, Eugene"https://zbmath.org/authors/?q=ai:bilokopytov.eugeneSummary: In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF's which only admit constant multipliers. To do that, using a method from
\textit{J.~Mashreghi} and \textit{T.~Ransford} [Anal. Math. Phys. 9, No.~2, 899--905 (2019; Zbl 1416.41005)], we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space of infinite cardinality. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.Isomorphisms from extremely regular subspaces of \(C_0(K)\) into \(C_0(S,X)\) spaceshttps://zbmath.org/1487.460252022-07-25T18:03:43.254055Z"Cerpa-Torres, Manuel Felipe"https://zbmath.org/authors/?q=ai:cerpa-torres.manuel-felipe"Rincón-Villamizar, Michael A."https://zbmath.org/authors/?q=ai:rincon-villamizar.michael-aSummary: For a locally compact Hausdorff space \(K\) and a Banach space \(X\), let \(C_0 (K,X)\) be the Banach space of all \(X\)-valued continuous functions defined on \(K\), which vanish at infinite provided with the sup norm. If \(X\) is \(\mathbb{R}\), we denote \(C_0 (K,X)\) as \(C_0 (K)\). If \(\mathcal{A} (K)\) be an extremely regular subspace of \(C_0(K)\) and \(T : \mathcal{A} (K) \longrightarrow C_0 (S,X)\) is an into isomorphism, what can be said about the set-theoretical or topological properties of \(K\) and \(S\)? Answering the question, we will prove that if \(X\) contains no copy of \(c_0\), then the cardinality of \(K\) is less than that of \(S\). Moreover, if \(\| T\| \| T^{-1}\| <3 \) and \(\mathcal{A}(K)\) is also a subalgebra of \(C_0 (K)\), the cardinality of the \(\alpha\) th derivative of \(K\) is less than that of the \(\alpha\) th derivative of \(S\), for each ordinal \(\alpha\).
Finally, if \(\lambda (X)>1\) and \(\| T\| \|T^{-1}\| <\lambda (X)\), then \(K\) is a continuous image of a subspace of \(S\). Here, \( \lambda \left(X\right)\) is the geometrical parameter introduced by \textit{K. Jarosz} [Pac. J. Math. 138, No. 2, 295--315 (1989; Zbl 0698.46033)]: \( \lambda (X)=\inf\{\max\{\| x+\lambda y\| :|\lambda|=1\}: \| x\| =\| y\| =1\}\). As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.Orthogonal elements in nonseparable rearrangement invariant spaceshttps://zbmath.org/1487.460262022-07-25T18:03:43.254055Z"Astashkin, S. V."https://zbmath.org/authors/?q=ai:astashkin.sergey-v"Semenov, E. M."https://zbmath.org/authors/?q=ai:semenov.evgenii-mSummary: Let \(E\) be a nonseparable rearrangement invariant space, and let \(E_0\) denote the closure of the set of all bounded functions in \(E\). We study elements of \(E\) orthogonal to the subspace \(E_0\), i.e., elements \(x \in E\) such that \({{\left\| x \right\|}_E} \leqslant{{\left\| {x + y} \right\|}_E}\) for any \(y \in{{E}_0} \).On Cesàro and Copson type function spaces. Reflexivityhttps://zbmath.org/1487.460312022-07-25T18:03:43.254055Z"Stepanov, Vladimir D."https://zbmath.org/authors/?q=ai:stepanov.vladimir-dmitrievichSummary: New ``weak''-type weighted Cesàro and Copson function spaces on the semi-axis are introduced. Unlike the standard Cesàro and Copson function spaces they possess associated reflexivity and their associate spaces are the weighted Sobolev spaces of the first order.On some properties of superreflexive Besov spaceshttps://zbmath.org/1487.460322022-07-25T18:03:43.254055Z"Agadzhanov, A. N."https://zbmath.org/authors/?q=ai:agadzhanov.a-nSummary: This paper contains results concerning superreflexive Besov spaces \(B_{p,q}^s(\mathbb{R}^n)\). Namely, expressions for convexity moduli and smoothness moduli with respect to the ``canonical'' norms are derived, and properties related to the finite representability of Banach spaces and linear compact operators in \(B_{p,q}^s(\mathbb{R}^n)\) are examined. Additionally, inequalities of the Prus-Smarzewski type for arbitrary equivalent norms and inequalities of the James-Gurariy type are presented. Based on the latter, two-sided estimates for the norms of elements in \(B_{p,q}^s(\mathbb{R}^n)\) can be obtained in terms of the expansion coefficients of these elements in unconditional normalized Schauder bases.Regularity of continuous multilinear operators and homogeneous polynomialshttps://zbmath.org/1487.460482022-07-25T18:03:43.254055Z"Kusraeva, Z. A."https://zbmath.org/authors/?q=ai:kusraeva.zalina-anatolevnaThe author extends to the case of multilinear operators and homogeneous polynomials the results about the regularity of continuous operators of [\textit{J.~Synnatzschke}, Vestn. Leningr. Univ., Mat. Mekh. Astron. 1972, No.~1, 60--69 (1972; Zbl 0234.47035); \textit{Y.~A. Abramovich} and \textit{A.~W. Wickstead}, Indag. Math., New Ser. 8, No.~3, 281--294 (1997; Zbl 0908.47031)].
The research utilizes the technique of the Fremlin tensor product.
Reviewer: S. S. Kutateladze (Novosibirsk)On Arens regularity of projective tensor product of Schatten spaceshttps://zbmath.org/1487.460522022-07-25T18:03:43.254055Z"Singh, Lav Kumar"https://zbmath.org/authors/?q=ai:singh.lav-kumarIn the context of normed algebras, Arens regularity means that the two Arens multiplications, defined by \textit{R.~Arens} [Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)], coincide. \textit{A.~Ülger} [Trans. Am. Math. Soc. 305, No.~2, 623--639 (1988; Zbl 0661.46046)], using bilinear forms on \(A \times B\), when \(A,B\) are Banach algebras, proved that the projective tensor product \(A \otimes^\gamma B\) is Arens regular if and only if any bilinear form on \(A \times B\) is biregular. In particular, he studied Arens regularity for certain Banach algebras, amongst them, for \(1< p<\infty\), \(\ell^p\) under the pointwise multiplication.
Based on this, the author turns to the noncommutative analogues of the \(\ell^p\) algebras. So, starting with a Hilbert space \(\mathcal{H}\), he takes the Banach algebra \(S_p(\mathcal{H})\) of all the Schatten \(p\)-class operators on \(\mathcal{H}\) with finite Schatten \(p\)-norm. In his research, the author presents various cases where the projective tensor product of certain Banach algebras fails to be Arens regular. For instance, when \(\mathcal{H}_1, \mathcal{H}_2\) are infinite dimensional Hilbert spaces, then the Banach algebra \(S_2(\mathcal{H}_1) \otimes^\gamma S_2(\mathcal{H}_2)\) is not Arens regular. In the final section of the paper, one of the main results is given, which answers the question for Arens regularity positively. More precisely, when the Hilbert space \(\mathcal{H}\) is separable and \(S_2(\mathcal{H})\) (the Banach algebra of Hilbert-Schmidt operators on \(\mathcal{H}\)) is equipped with the Schur product (defined appropriately via matrices, but unlike the matrix multiplication, it is a commutative one), then it becomes a completely continuous Banach algebra, and in turn, the projective tensor product \(S_2(\mathcal{H}) \otimes^\gamma S_2(\mathcal{H})\) is Arens regular.
Reviewer: Marina Haralampidou (Athína)On the completely bounded approximation property of crossed productshttps://zbmath.org/1487.460622022-07-25T18:03:43.254055Z"Meng, Qing"https://zbmath.org/authors/?q=ai:meng.qing.1|meng.qingSummary: Using the recently developed theory of Herz-Schur multipliers of a \(C^{\ast}\)-dynamical system, we prove equality of the Haagerup constants for a \(C^{\ast}\)-algebra and its crossed product by an amenable action.Isometries on noncommutative symmetric spaceshttps://zbmath.org/1487.460712022-07-25T18:03:43.254055Z"Sukochev, F. A."https://zbmath.org/authors/?q=ai:sukochev.fedor-a"Huang, J."https://zbmath.org/authors/?q=ai:huang.jinghaoA bounded linear operator \(T\) on a complex Banach space \(\mathcal X\) is said to be Hermitian if the numerical range \(\{\langle Tx,x \rangle: x\in\mathcal X\), \(\langle x,x\rangle=1\}\) of \(T\) is real, where \({\langle \cdot,\cdot\rangle }\) is a semi-inner product on \(\mathcal X\) consistent with the norm of \(\mathcal X\). Let \(\mathcal M\) be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace \(\tau\) (or else, an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space and let \(E(\mathcal M, \tau)\) be a separable symmetric space of \(\tau\)-measurable operators, whose norm is not proportional to the Hilbert norm \(\|\cdot\|_2\) on \(L^2(\mathcal M,\tau)\). The authors in this article show that a bounded operator \(T\) on \(E(\mathcal M, \tau)\) is Hermitian if and only if there exist selfadjoint operators \(A\) and \(B\) in \(\mathcal M\) such that \(Tx=Ax+xB\) for all \(x\in E(\mathcal M, \tau)\). In particular, \(T\) may be extended to a bounded Hermitian operator on the von Neumann algebra \(\mathcal M\).
Furthermore, they provide a description of all surjective linear isometries between these spaces. In fact, let \(E(\mathcal M_1,\tau_1)\) and \(E(\mathcal M_2, \tau_2)\) be two such spaces. If \(T\) is a surjective isometry from \(E(\mathcal M_1,\tau_1)\) onto \(E(\mathcal M_2, \tau_2)\), then there exist two sequences of operators: \(\{A_n: 1\leq n<\infty\}\subset E(\mathcal M_2,\tau_2)\) whose elements are pairwise disjointly supported from the right and \(\{B_n:1\leq n<\infty\}\subseteq E(\mathcal M_2,\tau_2)\) whose elements are pairwise disjointly supported from the left, and a surjective Jordan \(*\)-isomorphism \(J\) from \(\mathcal M_1\) onto \(\mathcal M_2\), and a central projection \(z\in \mathcal M_2\) such that, for any \(x\in E(\mathcal M_1,\tau_1)\cap \mathcal M_1\), \(T(x)=\sum_{n=1}^{\infty}(A_nJ(x)z+J(x)(I-z)B_n)\), where the series converges in the norm of the space \(E(\mathcal M_2,\tau_2)\).
Reviewer: Guoxing Ji (Xi'an)Subprojectivity of projective tensor products of Banach spaces of continuous functionshttps://zbmath.org/1487.460782022-07-25T18:03:43.254055Z"Causey, R. M."https://zbmath.org/authors/?q=ai:causey.ryan-michaelThe main result of [\textit{E. M. Galego} and \textit{C. Samuel}, Proc. Am. Math. Soc. 144, No.~6, 2611--2617 (2016; Zbl 1348.46011)] is extended. More precisely, it is proved that if \(n \in \mathbb N\) and \(K_1\),\dots , \(K_n\) are compact Hausdorff spaces, then the $n$-fold projective tensor product \(\widehat {\bigotimes}^{n}_{\pi, i=1} C(K_i)\) is \(c_0\)-saturated if and only if it is subprojective if and only if each \(K_i\) is scattered.
Reviewer: Elói M. Galego (São Paulo)Equivalence relations of projective operatorshttps://zbmath.org/1487.470032022-07-25T18:03:43.254055Z"Lin, Li Qiong"https://zbmath.org/authors/?q=ai:lin.liqiong"Lin, Hong Zhao"https://zbmath.org/authors/?q=ai:lin.hongzhao(no abstract)Some notes on projective operatorshttps://zbmath.org/1487.470042022-07-25T18:03:43.254055Z"Rong, Wei Jian"https://zbmath.org/authors/?q=ai:rong.weijian(no abstract)A toolkit for constructing dilations on Banach spaceshttps://zbmath.org/1487.470172022-07-25T18:03:43.254055Z"Fackler, Stephan"https://zbmath.org/authors/?q=ai:fackler.stephan"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochenSummary: We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if \(X\) is a super-reflexive Banach space and \(T\) is contained in the weakly closed convex hull of all invertible isometries on \(X\), then \(T\) admits a dilation to an invertible isometry on a Banach space \(Y\) with the same regularity as \(X\). The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.On infinitely singular operatorshttps://zbmath.org/1487.470332022-07-25T18:03:43.254055Z"Chen, Dong Xiao"https://zbmath.org/authors/?q=ai:chen.dongxiao"Chen, Jian Lan"https://zbmath.org/authors/?q=ai:chen.jianlan(no abstract)Singular irreducible \(M\)-operators on ordered Banach spaceshttps://zbmath.org/1487.470662022-07-25T18:03:43.254055Z"Kalauch, A."https://zbmath.org/authors/?q=ai:kalauch.anke"Lavanya, S."https://zbmath.org/authors/?q=ai:lavanya.s-r"Sivakumar, K. C."https://zbmath.org/authors/?q=ai:sivakumar.koratti-chengalrayanSummary: This work continues our earlier work [\textit{A. Kalauch} et al., Linear Algebra Appl. 565, 47--64 (2019; Zbl 1410.15059); Adv. Oper. Theory 4, No. 2, 481--496 (2019; Zbl 1454.47048)] on singular \(M\)-operators for operators over ordered Banach spaces, where we show how some interesting results on singular irreducible \(M\)-matrices have analogues to operators over ordered Banach spaces.A note on weak almost limited operatorshttps://zbmath.org/1487.470682022-07-25T18:03:43.254055Z"Machrafi, Nabil"https://zbmath.org/authors/?q=ai:machrafi.nabil"El Fahri, Kamal"https://zbmath.org/authors/?q=ai:el-fahri.kamal"Moussa, Mohammed"https://zbmath.org/authors/?q=ai:moussa.mohammed"Altin, Birol"https://zbmath.org/authors/?q=ai:altin.birolSummary: Let us recall that an operator \(T : E \rightarrow F\), between two Banach lattices, is said to be weak\(*\) Dunford-Pettis (resp., weak almost limited) if \(f_n(Tx_n) \rightarrow 0\) whenever \((x_n)\) converges weakly to 0 in \(E\) and \((f_n)\) converges weak\(*\) to 0 in \(F'\) (resp., \(f_n(Tx_n) \rightarrow 0\) for all weakly null sequences \((x_n) \subset E\) and all weak\(*\) null sequences \((f_n) \subset F'\) with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator \(R : G \rightarrow E\) (resp., \(S : F \rightarrow G\)), between Banach lattices, under which the product \(T R\) (resp., \(ST\)) is weak\(*\) Dunford-Pettis whenever \(T : E \rightarrow F\) is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp., their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.Coarse infinite-dimensionality of hyperspaces of finite subsetshttps://zbmath.org/1487.540202022-07-25T18:03:43.254055Z"Weighill, Thomas"https://zbmath.org/authors/?q=ai:weighill.thomas"Yamauchi, Takamitsu"https://zbmath.org/authors/?q=ai:yamauchi.takamitsu"Zava, Nicolò"https://zbmath.org/authors/?q=ai:zava.nicoloThe authors consider hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. They see that several coarse infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most \(n\) points. On the other hand, they prove that if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets contains a coarse disjoint union of any sequence of finite metric spaces (Theorem 5.2). As a corollary, such a hyperspace is not coarsely embeddable into any uniformly convex Banach space (Corollary 5.4). They also construct an unbounded 1-connected locally finite metric space that does not contain a sequence of long intervals coarsely and whose asymptotic dimension is equal to one (Example 6.1). In an appendix, they show that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has the metric sparsification property (Theorem 6.5).
Reviewer: Yutaka Iwamoto (Niihama)On Azuma-type inequalities for Banach space-valued martingaleshttps://zbmath.org/1487.600422022-07-25T18:03:43.254055Z"Luo, Sijie"https://zbmath.org/authors/?q=ai:luo.sijieSummary: In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space \(X\) is linearly isomorphic to a \(p\)-uniformly smooth space \((1<p\le 2)\) if and only if an Azuma-type inequality holds for \(X\)-valued martingales. This can be viewed as a generalization of Pinelis' work on an Azuma inequality for martingales with values in 2-uniformly smooth spaces. Secondly, an Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double indexed dyadic martingales and De la Peña-type inequalities for conditionally symmetric martingales, will also be discussed.