Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by O. Delgado and E.A. Sanchez Perez
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "Optimal domain of $q$-concave operators and vector measure representation of $q$-concave Banach lattices" by O. Delgado and E.A. Sanchez Perez. Abstract: Given a Banach space valued $q$-concave linear operator $T$ defined on a $\sigma$-order continuous quasi-Banach function space, we provide a description of the optimal domain of $T$ preserving $q$-concavity, that is, the largest $\sigma$-order continuous quasi-Banach function space to which $T$ can be extended as a $q$-concave operator. We show in this way the existence of maximal extensions for $q$-concave operators. As an application, we show a representation theorem for $q$-concave Banach lattices through spaces of integrable functions with respect to a vector measure. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years. Archive classification: math.FA Mathematics Subject Classification: 47B38, 46G10, 46E30, 46B42 Submitted from: easancpe(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.02337 or http://arXiv.org/abs/1511.02337

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Abstract of a paper by Jindrich Lechner
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "1-Grothendieck $C(K)$ spaces" by Jindrich Lechner. Abstract: A Banach space is said to be Grothendieck if weak and weak$^*$ convergent sequences in the dual space coincide. This notion has been quantificated by H. Bendov\'{a}. She has proved that $\ell_\infty$ has the quantitative Grothendieck property, namely, it is 1-Grothendieck. Our aim is to show that Banach spaces from a certain wider class are 1-Grothendieck, precisely, $C(K)$ is 1-Grothendieck provided $K$ is a totally disconnected compact space such that its algebra of clopen subsets has the so called Subsequential completeness property. Archive classification: math.FA Submitted from: jindrich.lechner(a)seznam.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.02202 or http://arXiv.org/abs/1511.02202

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Abstract of a paper by Tomasz Kania and Kent E. Morrison
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "The trace as an average over the unit sphere of a normed space with a 1-symmetric basis" by Tomasz Kania and Kent E. Morrison. Abstract: We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis. Our result is new even in the case of $\ell_p$-norms in $\mathbb{R}^N$ for $p\neq 2$. Archive classification: math.FA math.CA Mathematics Subject Classification: 15A60, 47A12 Submitted from: kmorriso(a)calpoly.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.02084 or http://arXiv.org/abs/1511.02084

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Abstract of a paper by David Alonso-Gutierrez, Bernardo Gonzalez Merino, Carlos Hugo Jimenez, and Rafael Villa
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "John's ellipsoid and the integral ratio of a log-concave function" by David Alonso-Gutierrez, Bernardo Gonzalez Merino, Carlos Hugo Jimenez, and Rafael Villa. Abstract: We extend the notion of John's ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional a?ne isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality. Archive classification: math.FA Submitted from: bg.merino(a)tum.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.01266 or http://arXiv.org/abs/1511.01266

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Abstract of a paper by Gonzalo Martinez-Cervantes
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "Riemann integrability versus weak continuity" by Gonzalo Martinez-Cervantes. Abstract: In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space $X$ is said to have the weak Lebesgue property if every Riemann integrable function from $[0,1]$ into $X$ is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under $\ell_1$-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford-Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of $\tau$-continuous non Riemann integrable functions, with $\tau$ a locally convex topology weaker than the norm topology. Archive classification: math.FA Mathematics Subject Classification: 46G10, 28B05, 03E10 Submitted from: gonzalo.martinez2(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.08801 or http://arXiv.org/abs/1510.08801

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Abstract of a paper by L. Garcia-Lirola, J. Orihuela, and M. Raja
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "Compact convex sets that admit a lower semicontinuous strictly convex function" by L. Garcia-Lirola, J. Orihuela, and M. Raja. Abstract: We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak$^*$ topology. In addition, we find exposed points where a strictly convex lower semicontinuous function is continuous. Archive classification: math.FA Mathematics Subject Classification: 46A55 (Primary) 46B03, 54E35 (Secondary) Remarks: 9 pages Submitted from: luiscarlos.garcia(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07921 or http://arXiv.org/abs/1510.07921

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Abstract of a paper by Miek Messerschmidt and Marten Wortel
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "The intrinsic metric on the unit sphere of a normed space" by Miek Messerschmidt and Marten Wortel. Abstract: Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary:46B10. Secondary: 51F99, 46B07 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07442 or http://arXiv.org/abs/1510.07442

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Abstract of a paper by Grigoris Paouris and Petros Valettas
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "On Dvoretzky's theorem for subspaces of $L_p$" by Grigoris Paouris and Petros Valettas. Abstract: We prove that for any $p > 2$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space $\ell_2^k$ can be $(1 + \varepsilon)$-embedded into $X$ with $k \geq c_p \min\{\varepsilon^2 n, (\varepsilon n)^{2/p} \}$, where $c_p > 0$ is a constant depending only on $p$. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 46B09 Remarks: 20 pages Submitted from: valettasp(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07289 or http://arXiv.org/abs/1510.07289

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Abstract of a paper by Grigoris Paouris, Petros Valettas and Joel Zinn
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "Random version of Dvoretzky's theorem in $\ell_p^n$" by Grigoris Paouris, Petros Valettas and Joel Zinn. Abstract: We study the dependence on $\varepsilon$ in the critical dimension $k(n, p, \varepsilon)$ that one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. For any fixed $n$ we give lower estimates for $k(n, p, \varepsilon)$ for all eligible values $p$ and $\varepsilon$, which agree with the sharp estimates for the extreme values $p = 1$ and $p = \infty$. In order to do so, we provide bounds for the gaussian concentration of the $\ell_p$-norm. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46B07, 46B09 Remarks: 45 pages Submitted from: valettasp(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07284 or http://arXiv.org/abs/1510.07284

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Abstract of a paper by Sofiya Ostrovska and Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Distortion in the finite determination result for embeddings of finite metric spaces into Banach spaces" by Sofiya Ostrovska and Mikhail I. Ostrovskii. Abstract: Given a Banach space $X$ and a locally finite metric space $A$, it is known that if all finite subsets of $A$ admit bilipschitz embeddings into $X$ with distortions $\le C$, then the space $A$ itself admits an embedding into $X$ with distortion $\le D\cdot C$, where $D$ is an absolute constant. The goal of this paper is to show that $D>1$, implying that, in general, there is a ``deterioration of distortion'' in the aforementioned situations. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B85, 46B20 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.05974 or http://arXiv.org/abs/1510.05974

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