Banach June 2008

banach@mathdept.okstate.edu

2 participants
10 discussions

Abstract of a paper by Vladimir Kadets, Varvara Shepelska and Dirk Werner
by alspach＠fourier.math.okstate.edu 18 Jun '08

18 Jun '08

This is an announcement for the paper "Quotients of Banach spaces with the Daugavet property" by Vladimir Kadets, Varvara Shepelska and Dirk Werner. Abstract: We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46B25; 47B38 Remarks: 15 pages The source file(s), dpry_bullpol_june08.tex: 55217 bytes, is(are) stored in gzipped form as 0806.1815.gz with size 17kb. The corresponding postcript file has gzipped size 114kb. Submitted from: werner(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.1815 or http://arXiv.org/abs/0806.1815 or by email in unzipped form by transmitting an empty message with subject line uget 0806.1815 or in gzipped form by using subject line get 0806.1815 to: math(a)arXiv.org.

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SUMIRFAS 2008
by Bill Johnson 17 Jun '08

17 Jun '08

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Abstract of a paper by Jaegil Kim and Han Ju Lee
by alspach＠fourier.math.okstate.edu 04 Jun '08

04 Jun '08

This is an announcement for the paper "Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices" by Jaegil Kim and Han Ju Lee. Abstract: Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset. In the last part, using Reisner's graph theoretic-approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\ell_\infty^n$. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm. Archive classification: math.FA math.CO Mathematics Subject Classification: 46G25; 46B20; 46B22; 52A21; 46B20 The source file(s), graph-June3-08.tex: 54865 bytes, is(are) stored in gzipped form as 0806.0507.gz with size 15kb. The corresponding postcript file has gzipped size 117kb. Submitted from: hahnju(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0507 or http://arXiv.org/abs/0806.0507 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0507 or in gzipped form by using subject line get 0806.0507 to: math(a)arXiv.org.

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Abstract of a paper by G. Androulakis, S. J. Dilworth, and N. J. Kalton
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces" by G. Androulakis, S. J. Dilworth, and N. J. Kalton. Abstract: We give a new approach to the Ramsey-type results of Gowers on block bases in Banach spaces and apply our results to prove the Gowers dichotomy in F-spaces. Archive classification: math.FA Mathematics Subject Classification: 46A16, 91A05, 91A80 The source file(s), AndDilKal.tex: 70671 bytes, is(are) stored in gzipped form as 0806.0058.gz with size 20kb. The corresponding postcript file has gzipped size 132kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0058 or http://arXiv.org/abs/0806.0058 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0058 or in gzipped form by using subject line get 0806.0058 to: math(a)arXiv.org.

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Abstract of a paper by G. Androulakis and K. Beanland
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators" by G. Androulakis and K. Beanland. Abstract: The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $\omega_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied. Archive classification: math.FA Mathematics Subject Classification: 47B07, 47A15 The source file(s), AlmostSC.tex: 41247 bytes, is(are) stored in gzipped form as 0806.0056.gz with size 12kb. The corresponding postcript file has gzipped size 95kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0056 or http://arXiv.org/abs/0806.0056 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0056 or in gzipped form by using subject line get 0806.0056 to: math(a)arXiv.org.

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Abstract of a paper by Hamed Hatami
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Graph norms and Sidorenko's conjecture" by Hamed Hatami. Abstract: Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for $A_G$, the adjacency matrix of a graph $G$, we have $h_H(A_G)=h_H(G)$. Let $m$ be the number of edges of $H$. It is easy to see that when $H$ is the cycle of length $2n$, then $h_H(\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann norm. We investigate a question of Lov\'{a}sz that asks for a characterization of graphs $H$ for which the function $h_H(\cdot)^{1/m}$ is a norm. We prove that $h_H(\cdot)^{1/m}$ is a norm if and only if a H\"{o}lder type inequality holds for $H$. We use this inequality to prove both positive and negative results, showing that $h_H(\cdot)^{1/m}$ is a norm for certain classes of graphs, and giving some necessary conditions on the structure of $H$ when $h_H(\cdot)^{1/m}$ is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the $h_H(\cdot)^{1/m}$ norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the $2n$-th Schatten-von Neumann norms. Archive classification: math.FA math.CO Mathematics Subject Classification: 46E30; 05C35 Remarks: to appear in Israel Journal of Mathematics The source file(s), arxiv/normFinal.bbl: 6949 bytes arxiv/normFinal.tex: 57888 bytes arxiv/normFinal.toc: 1082 bytes, is(are) stored in gzipped form as 0806.0047.tar.gz with size 20kb. The corresponding postcript file has gzipped size 125kb. Submitted from: hamed(a)cs.toronto.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0047 or http://arXiv.org/abs/0806.0047 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0047 or in gzipped form by using subject line get 0806.0047 to: math(a)arXiv.org.

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Abstract of a paper by Joaquim Martin and Mario Milman
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Isometry and symmetrization for logarithmic Sobolev inequalities" by Joaquim Martin and Mario Milman. Abstract: Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts. Archive classification: math.FA math.AP The source file(s), Gauss-final-rev.tex: 69485 bytes, is(are) stored in gzipped form as 0806.0021.gz with size 19kb. The corresponding postcript file has gzipped size 129kb. Submitted from: mario.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0806.0021 or http://arXiv.org/abs/0806.0021 or by email in unzipped form by transmitting an empty message with subject line uget 0806.0021 or in gzipped form by using subject line get 0806.0021 to: math(a)arXiv.org.

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Abstract of a paper by Markus Biegert
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Lattice homomorphisms between Sobolev spaces" by Markus Biegert. Abstract: We show that every vector lattice homomorphism $T$ from $W^{1,p}_0(\Omega_1)$ into $W^{1,q}(\Omega_2)$ for $p,q\in (1,\infty)$ and open sets \Omega_1,\Omega_2\subset\IR^N$ has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\Omega_1$ and $g:\Omega_2\to[0,\infty)$. This representation follows as an application of an abstract and more general representation theorem, which can be applied in many other situations. We prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $W^{1,q}(\Omega_2)$ admits a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\overline\Omega_1$ and $g:\Omega_2\to[0,\infty)$. Here $\tsW^{1,p}(\Omega_1)$ denotes the closure of $W^{1,p}(\Omega_1)\cap C_c(\overline\Omega_1)$ in $W^{1,p}(\Omega_1)$ and every $u\in\tsW^{1,p}(\Omega_1)$ admits a trace on the boundary $\partial\Omega_1$ of $\Omega_1$. Finally we prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $\tsW^{1,q}(\Omega_2)$ where $\Omega_1$ is bounded has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_{q,\Omega_2}$-quasi everywhere on }\overline\Omega_2$ with mappings $\xi:\overline\Omega_2\to\overline\Omega_1$ and $g:\overline\Omega_2\to[0,\infty)$. At the end of this article we consider also lattice isomorphisms between Sobolev spaces and the representation of their inverses. Archive classification: math.AP math.FA The source file(s), orderhomomorphism.bbl: 4468 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0805.4740 or http://arXiv.org/abs/0805.4740 or by email in unzipped form by transmitting an empty message with subject line uget 0805.4740 or in gzipped form by using subject line get 0805.4740 to: math(a)arXiv.org.

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Abstract of a paper by D. Drivaliaris and N. Yannakakis
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Hilbert space structure and positive operators" by D. Drivaliaris and N. Yannakakis. Abstract: Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case. Archive classification: math.FA Mathematics Subject Classification: 46B03; 46C15; 47B99 Citation: Journal of Mathematical Analysis and Applications 305 (2) (2005), The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0805.4721 or http://arXiv.org/abs/0805.4721 or by email in unzipped form by transmitting an empty message with subject line uget 0805.4721 or in gzipped form by using subject line get 0805.4721 to: math(a)arXiv.org.

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Abstract of a paper by D. Drivaliaris and N. Yannakakis
by alspach＠fourier.math.okstate.edu 03 Jun '08

03 Jun '08

This is an announcement for the paper "Subspaces with a common complement in a Banach space" by D. Drivaliaris and N. Yannakakis. Abstract: We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution S on X exchanging the two subspaces, such that I+S is bounded from below on their union. Moreover we show that, in a separable Hilbert space, the only pairs of subspaces with a common complement are those which are either equivalently positioned or not completely asymptotic to one another. We also obtain characterizations for the existence of a common complement for subspaces with closed sum. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46C05; 47A05 Citation: Studia Mathematica 182 (2) (2007), 141-164 The source file(s), common_arxiv.tex: 74237 bytes, is(are) stored in gzipped form as 0805.4707.gz with size 16kb. The corresponding postcript file has gzipped size 104kb. Submitted from: d.drivaliaris(a)fme.aegean.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0805.4707 or http://arXiv.org/abs/0805.4707 or by email in unzipped form by transmitting an empty message with subject line uget 0805.4707 or in gzipped form by using subject line get 0805.4707 to: math(a)arXiv.org.

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Banach June 2008