Banach September 2018

banach@mathdept.okstate.edu

2 participants
23 discussions

Abstract of a paper by György Pál Gehér, Tamás Titkos, Dániel Virosztek
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On isometric embeddings of Wasserstein spaces -- the discrete case” by György Pál Gehér<https://arxiv.org/search/math?searchtype=author&query=Geh%C3%A9r%2C+G+P>, Tamás Titkos<https://arxiv.org/search/math?searchtype=author&query=Titkos%2C+T>, Dániel Virosztek<https://arxiv.org/search/math?searchtype=author&query=Virosztek%2C+D>. Abstract: The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\mathcal{W}_p(\mathcal{X})$ splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that $\mathcal{W}_p(\mathcal{X})$ is isometrically rigid for all $0<p<\infty$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.01101

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Abstract of a paper by Joscha Prochno, Christoph Thäle, Nicola Turchi
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Geometry of $\ell_p^n$-balls: Classical results and recent developments” by Joscha Prochno<https://arxiv.org/search/math?searchtype=author&query=Prochno%2C+J>, Christoph Thäle<https://arxiv.org/search/math?searchtype=author&query=Th%C3%A4le%2C+C>, Nicola Turchi<https://arxiv.org/search/math?searchtype=author&query=Turchi%2C+N>. Abstract: This survey will appear as a chapter in the forthcoming "Proceedings of the HDP VIII Conference". The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.10435

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Abstract of a paper by Julio Becerra-Guerrero, María Cueto-Avellaneda, Francisco J. Fernández-Polo, Antonio M. Peralta
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space” by Julio Becerra-Guerrero<https://arxiv.org/search/math?searchtype=author&query=Becerra-Guerrero%2C+J>, María Cueto-Avellaneda<https://arxiv.org/search/math?searchtype=author&query=Cueto-Avellaneda%2C+M>, Francisco J. Fernández-Polo<https://arxiv.org/search/math?searchtype=author&query=Fern%C3%A1ndez-Polo%2…>, Antonio M. Peralta<https://arxiv.org/search/math?searchtype=author&query=Peralta%2C+A+M>. Abstract: We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.01460

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Abstract of a paper by Made Tantrawan, Denny H. Leung
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On closedness of convex sets in Banach function spaces” by Made Tantrawan<https://arxiv.org/search/math?searchtype=author&query=Tantrawan%2C+M>, Denny H. Leung<https://arxiv.org/search/math?searchtype=author&query=Leung%2C+D+H>. Abstract: Let $\mathcal{X}$ be a Banach function space. A well-known problem arising from theory of risk measures asks when the order closedness of a convex set in $\mathcal{X}$ implies the closedness with respect to the topology $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$ where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. In this paper, we give an answer to the problem for a large class of Banach function spaces. We show that under the Fatou property and the subsequence splitting property, every order closed convex set in $\mathcal{X}$ is $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closed if and only if either $\mathcal{X}$ or the norm dual $\mathcal{X}^*$ of $\mathcal{X}$ is order continuous. In addition, we also give a characterization of $\mathcal{X}$ for which the order closedness of a convex set in $\mathcal{X}$ is equivalent to the closedness with respect to the topology $\sigma(\mathcal{X},\mathcal{X}_{uo}^\sim)$ where $\mathcal{X}_{uo}^\sim$ is the unbounded order continuous dual of $\mathcal{X}$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.06747

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Abstract of a paper by B. P. Duggal
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Isolated eigenvalues, poles and compact perturbations of Banach space operators” by B. P. Duggal<https://arxiv.org/search/math?searchtype=author&query=Duggal%2C+B+P>. Abstract: Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $\Pi(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $\Pi^a(A)$) of the spectrum of $A$ satisfy $\Pi(A)\subseteq E(A)$ (resp., $\Pi^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $\Pi(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $\Pi^a(A)$. The study of identities $E(A)=\Pi^a(A)$ and $E^a(A)=\Pi(A)$, and their stability under perturbation by commuting Riesz operators, has been of some interest in the recent past. This paper studies the stability of these identities under perturbation by (non-commuting) compact operators. Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.03542

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Abstract of a paper by Bruno de Mendonça Braga
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On asymptotically uniformly smoothness and nonlinear geometry of Banach spaces” by Bruno de Mendonça Braga<https://arxiv.org/search/math?searchtype=author&query=de+Mendon%C3%A7a+Brag…>. Abstract: These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.03254

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Abstract of a paper by Jesús M. F. Castillo, Ricardo García
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Bilinear forms and the $\Ext^2$-problem in Banach spaces” by Jesús M. F. Castillo<https://arxiv.org/search/math?searchtype=author&query=Castillo%2C+J+M+F>, Ricardo García<https://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa%2C+R>. Abstract: Let $X$ be a Banach space and let $\kappa(X)$ denote the kernel of a quotient map $\ell_1(\Gamma)\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if bilinear forms on $\kappa(X)$ extend to $\ell_1(\Gamma)$. From that we obtain i) If $\kappa(X)$ is a $\mathcal L_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$ is separable, $\kappa(X)$ is not an $\mathcal L_1$ space and $\Ext^2(X,X^*)=0$ then $\kappa(X)$ has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces. https://arxiv.org/abs/1808.03173

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Abstract of a paper by Félix Cabello Sánchez, Jesús M. F. Castillo, Yolanda Moreno
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On the bounded approximation property on subspaces of $\ell_p$ when $0<p<1$ and related issues” by Félix Cabello Sánchez<https://arxiv.org/search/math?searchtype=author&query=S%C3%A1nchez%2C+F+C>, Jesús M. F. Castillo<https://arxiv.org/search/math?searchtype=author&query=Castillo%2C+J+M+F>, Yolanda Moreno<https://arxiv.org/search/math?searchtype=author&query=Moreno%2C+Y>. Abstract: This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec-Pe\l czy\'nski-Wojtaszczyk complementably universal spaces for Banach spaces with the BAP. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.03169

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Abstract of a paper by Andrés Felipe Muñoz Tello
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Gâteaux differentiability on non-separable Banach spaces” by Andrés Felipe Muñoz Tello<https://arxiv.org/search/math?searchtype=author&query=Tello%2C+A+F+M>. Abstract: This paper deals with the extension of a classical theorem by R. Phelps on the G\^ateaux differentiability of Lipschitz functions on separable Banach spaces to the non-separable case. The extension of the theorem is not possible for general non-separable Banach spaces, as shown by Larman. Therefore, we will work on some important non-separable spaces. We consider a norm of the non-separable space $\ell^{\infty}(\mathbb{R})$, showing that it complies with the aforementioned theorem thesis. We also consider other examples as $L^{\infty}(\mathbb{R})$ and $NBV[a,b]$, showing in all cases their sets of $G$-differentiability and some properties of these sets. All of the above, closely following the assumptions in Phelps Theorem, which will extend in the case separable to functions with rank over the Asplund spaces and in the non-separable case for projective limits and locally-Lipschitz cylindrical functions. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.02843

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Abstract of a paper by Debmalya Sain, Anubhab Ray, Kallol Paul
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Extreme contractions on finite-dimensional polygonal Banach spaces” by Debmalya Sain<https://arxiv.org/search/math?searchtype=author&query=Sain%2C+D>, Anubhab Ray<https://arxiv.org/search/math?searchtype=author&query=Ray%2C+A>, Kallol Paul<https://arxiv.org/search/math?searchtype=author&query=Paul%2C+K>. Abstract: We explore extreme contractions between finite-dimensional polygonal Banach spaces, from the point of view of attainment of norm of a linear operator. We prove that if $ X $ is an $ n- $dimensional polygonal Banach space and $ Y $ is any Banach space and $ T \in L(X,Y) $ is an extreme contraction, then $ T $ attains norm at $ n $ linearly independent extreme points of $ B_{X}. $ Moreover, if $ T $ attains norm at exactly $ n $ linearly independent extreme points $ x_1, x_2, \ldots, x_n $ of $ B_X $ and does not attain norm at any other extreme point of $ B_X, $ then each $ Tx_i $ is an extreme point of $ B_Y.$ We completely characterize extreme contractions between a finite-dimensional polygonal Banach space and a strictly convex Banach space. We introduce L-P property for a pair of Banach spaces and show that it has natural connections with our present study. We also prove that for any strictly convex Banach space $ X $ and any finite-dimensional polygonal Banach space $ Y, $ the pair $ (X,Y) $ does not have L-P property. Finally, we obtain a characterization of Hilbert spaces among strictly convex Banach spaces in terms of L-P property. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.01881

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Banach September 2018