Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Guillermo P. Curbera and Werner J. Ricker
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "The weak Banach-Saks property for function spaces" by Guillermo P. Curbera and Werner J. Ricker. Abstract: We establish the weak Banach-Saks property for function spaces arising as the optimal domain of an operator. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46G10 Submitted from: curbera(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.05728 or http://arXiv.org/abs/1512.05728

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Abstract of a paper by Jesus A. Jaramillo, Raquel Gonzalo and Diego Yanez
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Asymptotic Smoothness, Convex Envelopes and Polynomial Norms" by Jesus A. Jaramillo, Raquel Gonzalo and Diego Yanez. Abstract: We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N. Archive classification: math.FA Submitted from: jaramil(a)mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.05407 or http://arXiv.org/abs/1512.05407

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Abstract of a paper by Paul F.X. Muller and Johanna Penteker
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces" by Paul F.X. Muller and Johanna Penteker. Abstract: For $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(\delta^2), \quad (\varphi_{I\times J}) \mapsto \sum \varphi_{I\times J}f_{I \times J}h_{I \times J}. \] Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(\delta^2)$ \[|f|=|x|^{1-\theta}|y|^{\theta}\quad\quad \text{ and }\quad\quad \|x\|_{X_0}^{1-\theta}\|y\|^{\theta}_{H^2(\delta^2)}\leq C\|f\|_{H^p(\delta^2)}, \] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(\delta^2)$ and $H^2(\delta^2)$. Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_f$ involves the Haar coefficients of $f$ and its atomic decomposition. We treated the one-parameter $H^p$-spaces in [P.F.X M\"uller, J.Penteker, $p$-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, Houston Journal Math.,41(2):639-668,2015.]. Archive classification: math.FA Mathematics Subject Classification: 42B30 46B25 46B09 46B42 46E40 47B10 60G42 Remarks: 10 pages Submitted from: johanna.penteker(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.04790 or http://arXiv.org/abs/1512.04790

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Abstract of a paper by Geraldo Botelho, Jamilson R. Campos, and Joedson Santos
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Operator ideals related to absolutely summing and Cohen strongly operators" by Geraldo Botelho, Jamilson R. Campos, and Joedson Santos. Abstract: We study the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha \cite{karnsinha} and the classical spaces of absolutely, weakly and Cohen strongly summable sequences. As applications, we prove a new factorization theorem for absolutely summing operators and a contribution to the existence of infinite dimensional spaces formed by non-absolutely summing operators is given. Archive classification: math.FA Mathematics Subject Classification: 46B45, 47B10, 47L20 Remarks: 15 pages Submitted from: jamilson(a)dce.ufpb.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.04713 or http://arXiv.org/abs/1512.04713

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Abstract of a paper by Aicke Hinrichs, Anton Kolleck, and Jan Vybiral
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Carl's inequality for quasi-Banach spaces" by Aicke Hinrichs, Anton Kolleck, and Jan Vybiral. Abstract: We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $\alpha>0$ there exists a constant $c_\alpha>0$ such that $$ \sup_{1\le k\le n}k^{\alpha}e_k(T)\le c_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfand number of $T$. For Banach spaces $X$ and $Y$ this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result, which closes a gap in the argument of Donoho in his seminal paper on compressed sensing. Archive classification: math.FA Remarks: 12 pages Submitted from: aicke.hinrichs(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.04421 or http://arXiv.org/abs/1512.04421

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez, and Abraham Rueda
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Octahedrality in Lipschitz free Banach spaces" by Julio Becerra Guerrero, Gines Lopez-Perez, and Abraham Rueda. Abstract: The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it. As a consequence, we get that the space of Lipschitz functions on a metric space valued in a dual Banach space satisfies the weak-star strong diameter two property, under natural topological hipothesess on the metric space. Also, we show an example proving that these hypotheses are optimal. Archive classification: math.FA Remarks: 18 pages Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.03558 or http://arXiv.org/abs/1512.03558

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Abstract of a paper by Sergey V. Astashkin, Karol Lesnik, and Lech Maligranda
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Isomorphic structure of Ces\`aro and Tandori spaces" by Sergey V. Astashkin, Karol Lesnik, and Lech Maligranda. Abstract: We investigate the isomorphic structure of the Ces\`aro spaces and their duals, the Tandori spaces. The main result states that the Ces\`aro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}} =\|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t): \esssup_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Ces\`aro and Tandori spaces. In particular, using Bourgain's results we show that a wide class of Ces{\`a}ro-Marcinkiewicz and Ces{\`a}ro-Lorentz spaces have the latter property. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B42 Submitted from: lech.maligranda(a)ltu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.03336 or http://arXiv.org/abs/1512.03336

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Abstract of a paper by Bruno de Mendonca Braga
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Coarse and uniform embeddings" by Bruno de Mendonca Braga. Abstract: In these notes, we study the relation between uniform and coarse embed= dings between Banach spaces. In order to understand this relation better, we al= so look at the problem of when a coarse embedding can be assumed to be topological. Among other results, we show that if a Banach space $X$ uniformly embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and uniformly embeds into $Y$, and if a Banach space $X$ coarsely embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and homeomorphically embeds into $Y$ by a map with uniformly continuous inverse. Archive classification: math.FA Mathematics Subject Classification: 46B80 Submitted from: demendoncabraga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.03109 or http://arXiv.org/abs/1512.03109

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Abstract of a paper by Guillermo P. Curbera and Werner J. Ricker
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Abstract Ces\`aro spaces: Integral representations" by Guillermo P. Curbera and Werner J. Ricker. Abstract: The Ces\`aro function spaces $Ces_p=[C,L^p]$, $1\le p\le\infty$, have received renewed attention in recent years. Many properties of $[C,L^p]$ are known. Less is known about $[C,X]$ when the Ces\`aro operator takes its values in a rearrangement invariant (r.i.) space $X$ other than $L^p$. In this paper we study the spaces $[C,X]$ via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of $[C,X]$ and the Fatou completion of $[C,X]$; to show that $[C,X]$ is never reflexive and never r.i.; to identify when $[C,X]$ is weakly sequentially complete, when it is isomorphic to an AL-space, and when it has the Dunford-Pettis property. The same techniques are used to analyze the operator $C:[C,X]\to X$; it is never compact but, it can be completely continuous. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46G10 Remarks: 21 pages Submitted from: curbera(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.02760 or http://arXiv.org/abs/1512.02760

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Abstract of a paper by Mathieu Meyer and Shlomo Reisner
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "The isotropy constant and boundary properties of convex bodies" by Mathieu Meyer and Shlomo Reisner. Abstract: Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endo= wed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has posit= ive generalized Gauss curvature at some point of its boundary, then $K$ is no= t a local maximizer for the isotropy constant $L_K$. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20, 52A20, 53A05 Submitted from: reisner(a)math.haifa.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.02927 or http://arXiv.org/abs/1512.02927

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