Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Florent P. Baudier, Robert Deville
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Lipschitz Embeddings of Metric Spaces into $c_0$” by Florent P. Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Robert Deville<https://arxiv.org/find/math/1/au:+Deville_R/0/1/0/all/0/1>. Abstract: Let M be a separable metric space. We say that $f=(f_n): M\rightarrow c_0$ is a good-$\lambda$-embedding if, whenever $x, y\in M, x\neq y$ implies $d(x, y)\leq \|f(x)-f(y)\|$ and, for each $n, Lip(f_n)<\lambfda$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$\lambda$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $\pi(\lambda)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.02025

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Abstract of a paper by Atanu Manna
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Certain geometric structure of Λ-sequence spaces” by Atanu Manna<https://arxiv.org/find/math/1/au:+Manna_A/0/1/0/all/0/1>. Abstract: The $Lambda$-sequence spaces $\Lambda_p$ for $1<p\leq\infty$ and its generalization $\Lambda_{\hat{p}}$ for $1<\hat{p}<\infty, \hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $\ell_{\hat{p}}(R_{n+1})$ of finite dimensional Euclidean space $R_{n+1}$. Hence it follows that sequence spaces $\Lambda_{p}$ and $\Lambda_{\hat{p}}$ possesses the uniform Opial property, property ($\beta$) of Rolewicz and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec-Klee property. Further necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ is being carried out. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01519

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Abstract of a paper by Aude Dalet, Gilles Lancien
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Some properties of coarse Lipschitz maps between Banach spaces” by Aude Dalet<https://arxiv.org/find/math/1/au:+Dalet_A/0/1/0/all/0/1>, Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>. Abstract: We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01364

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Abstract of a paper by Dan Freeman, Thomas Schlumprecht, Andras Zsak
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Closed Ideals of Operators between the Classical Sequence Spaces” by Dan Freeman<https://arxiv.org/find/math/1/au:+Freeman_D/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Andras Zsak<https://arxiv.org/find/math/1/au:+Zsak_A/0/1/0/all/0/1>. Abstract: We prove that the spaces $\mathcal{L}(\ell_p, c_0), \mathcal{L}(\ell_p, \ell_{\inty})$ and $\mathcal{L}(\ell_1, \ell_q)$ of operators with $1<p, q<\infty$ have continuum many closed ideals. This extends and improves earlier works by Schlumprecht and Zs\'ak, by Wallis, and by Sirotkin and Wallis. Several open problems remain. Key to our construction of closed ideals are matrices with the Restricted Isometry Property that come from Compressed Sensing. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01153

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Abstract of a paper by Ben Wallis
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Garling sequence spaces” by Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: By generalizing a construction of Garling, for each $1\leq p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0-\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta})_{n=1}^{\infty}$ for some $0<\theta<1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01145

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Abstract of a paper by Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Daugavet property and separability in Banach spaces” by Abraham Rueda Zoca<https://arxiv.org/find/math/1/au:+Zoca_A/0/1/0/all/0/1>. Abstract: We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former characterisation also motivates the introduction of two indices related to the Daugavet property and a short study of them.. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.09698

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Abstract of a paper by Torrey M. Gallagher
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “A weak convergence theorem for mean nonexpansive mappings” by Torrey M. Gallagher<https://arxiv.org/find/math/1/au:+Gallagher_T/0/1/0/all/0/1>. Abstract: In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces satisfying Opial's condition. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.09390

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Abstract of a paper by Miek Messerschmidt
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “A Pointwise Lipschitz Selection Theorem $” by Miek Messerschmidt<https://arxiv.org/find/math/1/au:+Messerschmidt_M/0/1/0/all/0/1>. Abstract: We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.08435

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Abstract of a paper by J.M. Calabuig, J. Rodríguez, P. Rueda, E.A. Sánchez-Pérez
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On p-Dunford integrable functions with values in Banach spaces” by J.M. Calabuig<https://arxiv.org/find/math/1/au:+Calabuig_J/0/1/0/all/0/1>, J. Rodríguez<https://arxiv.org/find/math/1/au:+Rodriguez_J/0/1/0/all/0/1>, P. Rueda<https://arxiv.org/find/math/1/au:+Rueda_P/0/1/0/all/0/1>, E.A. Sánchez-Pérez<https://arxiv.org/find/math/1/au:+Sanchez_Perez_E/0/1/0/all/0/1>. Abstract: Let $(\Omega, \Sigma, \mu)$ be a complete probability space, $X$ a Banach space and $1\leq p<\infty$. In this paper we discuss several aspects of $p$-Dunford integrable functions $f: \Omega\rightarrow X$. Special attention is paid to the compactness of the Dunford operator of $f$. We also study the $p$-Bochner integrability of the composition $u\circ f: \Omega\rightarrow Y$, where $u$ is a $p$-summing operator from $X$ to another Banach space $Y$. Finally, we also provide some tests of $p$-Dunford integrability by using $w^*$-thick subsets of $X^*$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.08087

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Abstract of a paper by W. Cuellar-Carrera
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Non-ergodic Banach spaces are near Hilbert” by W. Cuellar-Carrera<https://arxiv.org/find/math/1/au:+Cuellar_Carrera_W/0/1/0/all/0/1>. Abstract: We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ $(2<p<\infty)$ is ergodic. This reinforces the conjecture that $\el_2$ is the only non ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p<2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.05500

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