November 2017 - Banach - mathdept.okstate.edu

Abstract of a paper by David Alonso-Gutierrez, Joscha Prochno, Christoph Thaele
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls” by David Alonso-Gutierrez<https://arxiv.org/find/math/1/au:+Alonso_Gutierrez_D/0/1/0/all/0/1>, Joscha Prochno<https://arxiv.org/find/math/1/au:+Prochno_J/0/1/0/all/0/1>, Christoph Thaele<https://arxiv.org/find/math/1/au:+Thaele_C/0/1/0/all/0/1>. Abstract: In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any n∈ℕ let En be a random subspace of dimension $k_n\in\{1,…,n\}$ and $X_n$ be a random point in the unit ball of $\ell_p^n$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_n}X_n\|_2$ of random orthogonal projections of $X_n$ onto $E_n$. In particular, under the condition that $k_n\rightarrow\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\rightarrow\infty$ fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end we provide a discussion of the large deviations counterpart to our central limit theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.10130

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Abstract of a paper by Ian Doust, Shaymaa Al-shakarchi
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Isomorphisms of $AC(\sigma)$ spaces for countable sets” by Ian Doust<https://arxiv.org/find/math/1/au:+Doust_I/0/1/0/all/0/1>, Shaymaa Al-shakarchi<https://arxiv.org/find/math/1/au:+Al_shakarchi_S/0/1/0/all/0/1>. Abstract: It is known that the classical Banach--Stone theorem does not extend to the class of $AC(\sigma)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $\sigma$ is restricted to the set of compact polygons, then all the corresponding $AC(\sigma)$ spaces are isomorphic. In this paper we examine the case where $\sigma$ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.09073

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Abstract of a paper by Bruno de Mendonça Braga
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Nonlinear weakly sequentially continuous embeddings between Banach spaces” by Bruno de Mendonça Braga<https://arxiv.org/find/math/1/au:+Braga_B/0/1/0/all/0/1>. Abstract: In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $$\|e_1+…+e_k\|_{\bar{\delta_Y}\leq\|e_1+…+e_k\|_S$$, where $\bar{\delta_Y}$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.07852

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Abstract of a paper by Ryan M. Causey, Gilles Lancien
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Prescribed Szlenk index of iterated duals” by Ryan M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>. Abstract: In a previous work, the first named author described the set $ \mathbb{P}$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\mathbb{P}$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in $\{0,…, n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.01638

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Abstract of a paper by Andrew Swift
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “A coding of bundle graphs and their embeddings into Banach spaces” by Andrew Swift<https://arxiv.org/find/math/1/au:+Swift_A/0/1/0/all/0/1>. Abstract: The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontrivial finitely-branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontrivial $\mathcal{N}_0$-branching bundle graph. The best known distortions are recovered. For the specific case of $L_1$, it is shown that every countably-branching bundle graph bi-Lipschitzly embeds into $L_1$ with distortion no worse than $2$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.00877

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