Banach October 2009

banach@mathdept.okstate.edu

1 participants
18 discussions

Abstract of a paper by Manor Mendel and Assaf Naor
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Towards a calculus for non-linear spectral gaps [extended abstract]" by Manor Mendel and Assaf Naor. Abstract: Given a finite regular graph G=(V,E) and a metric space (X,d_X), let $gamma_+(G,X) denote the smallest constant $\gamma_+>0$ such that for all f,g:V\to X we have: \frac{1}{|V|^2}\sum_{x,y\in V} d_X(f(x),g(y))^2\le \frac{\gamma_+}{|E|} \sum_{xy\in E} d_X(f(x),g(y))^2. In the special case X=R this quantity coincides with the reciprocal of the absolute spectral gap of $G$, but for other geometries the parameter \gamma_+(G,X), which we still think of as measuring the non-linear spectral gap of G with respect to X (even though there is no actual spectrum present here), can behave very differently. Non-linear spectral gaps arise often in the theory of metric embeddings, and in the present paper we systematically study the theory of non-linear spectral gaps, partially in order to obtain a combinatorial construction of super-expander -- a family of bounded-degree graphs G_i=(V_i,E_i), with \lim_{i\to \infty} |V_i|=\infty, which do not admit a coarse embedding into any uniformly convex normed space. In addition, the bi-Lipschitz distortion of G_i in any uniformly convex Banach space is \Omega(\log |V_i|), which is the worst possible behavior due to Bourgain's embedding theorem. Such remarkable graph families were previously known to exist due to a tour de force algebraic construction of Lafforgue. Our construction is different and combinatorial, relying on the zigzag product of Reingold-Vadhan-Wigderson. Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: 51F99, 05C12, 05C50, 46B85 Remarks: 32 pages. Extended abstract. To be published (in abridged form) in the proceedings of the ACM-SIAM Symposium on Discrete Algorithms 2010 (SODA '10) The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: mendelma(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.2041 or http://arXiv.org/abs/0910.2041 or by email in unzipped form by transmitting an empty message with subject line uget 0910.2041 or in gzipped form by using subject line get 0910.2041 to: math(a)arXiv.org.

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Abstract of a paper by Necip Simsek, Ekrem Savas, and Vatan Karakaya
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Some geometric and topological properties of a new sequence space defined by De la Vallee-Poussin mean" by Necip Simsek, Ekrem Savas, and Vatan Karakaya. Abstract: The main purpose of this paper is to introduce a new sequence space by using de la Vallee-Poussin mean and investigate both the modular structure with some geometric properties and some topological properties with respect to the Luxemburg norm. Archive classification: math.FA Mathematics Subject Classification: 46A45, 46B20, 46B45 (Primary) Remarks: 12 pages The source file(s), POISON.tex: 37921 bytes, is(are) stored in gzipped form as 0910.1947.gz with size 10kb. The corresponding postcript file has gzipped size 88kb. Submitted from: nsimsek(a)adiyaman.edu.tr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.1947 or http://arXiv.org/abs/0910.1947 or by email in unzipped form by transmitting an empty message with subject line uget 0910.1947 or in gzipped form by using subject line get 0910.1947 to: math(a)arXiv.org.

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Abstract of a paper by Wieslaw Kubis
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Finitely fibered Rosenthal compacta and trees" by Wieslaw Kubis. Abstract: We study some topological properties of trees with the interval topology. In particular, we characterize trees which admit a $2$-fibered compactification and we present two examples of trees whose one-point compactifications are Rosenthal compact with certain renorming properties of their spaces of continuous functions. Archive classification: math.GN math.FA Mathematics Subject Classification: 54D30, 46B03, 46E15, 54C35, 54G12. Remarks: 16 pages The source file(s), small_noK_ver4a.tex: 58405 bytes, is(are) stored in gzipped form as 0910.1360.gz with size 18kb. The corresponding postcript file has gzipped size 110kb. Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.1360 or http://arXiv.org/abs/0910.1360 or by email in unzipped form by transmitting an empty message with subject line uget 0910.1360 or in gzipped form by using subject line get 0910.1360 to: math(a)arXiv.org.

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Abstract of a paper by R.D. Arthan
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Aronszajn's criterion for Euclidean space" by R.D. Arthan. Abstract: We give a simple proof of a characterization of euclidean space due to Aronszajn and derive a well-known characterization due to Jordan & von Neumann as a corollary. Archive classification: math.GM math.FA Mathematics Subject Classification: 46B20; 46C05 Remarks: 1 figure The source file(s), 73.bbl: 714 bytes 73.tex: 17428 bytes 73a.eps: 11329 bytes, is(are) stored in gzipped form as 0910.0608.tar.gz with size 9kb. The corresponding postcript file has gzipped size 30kb. Submitted from: rda(a)lemma-one.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.0608 or http://arXiv.org/abs/0910.0608 or by email in unzipped form by transmitting an empty message with subject line uget 0910.0608 or in gzipped form by using subject line get 0910.0608 to: math(a)arXiv.org.

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Abstract of a paper by Imre Patyi
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "On holomorphic domination, I" by Imre Patyi. Abstract: Let $X$ be a separable Banach space and $u{:}\,X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:}\,X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_1[0,1]$ of summable functions, then there is a $C^1$-smooth function $u$ on $X$ with $\overline\partial u=f$ on $X$. Archive classification: math.CV math.FA Mathematics Subject Classification: 32U05; 32L10; 46G20 The source file(s), holodom-I-3.tex: 35922 bytes, is(are) stored in gzipped form as 0910.0476.gz with size 12kb. The corresponding postcript file has gzipped size 82kb. Submitted from: i355p113(a)speedpost.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.0476 or http://arXiv.org/abs/0910.0476 or by email in unzipped form by transmitting an empty message with subject line uget 0910.0476 or in gzipped form by using subject line get 0910.0476 to: math(a)arXiv.org.

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Abstract of a paper by Abdellatif Bourhim
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Additive maps preserving the reduced minimum modulus of Banach space operators" by Abdellatif Bourhim. Abstract: Let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $\varphi$ on ${\mathcal B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\to X$ and $V:X\to X$ both linear or both conjugate linear such that $\varphi(T)=UTV$ for all $T\in{\mathcal B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\to X$ and $V:X\to X^*$ both linear or both conjugate linear such that $\varphi(T)=UT^*V$ for all $T\in{\mathcal B}(X)$. As immediate consequences of the ingredients used in the proof of this result, we get the complete description of surjective additive maps preserving the minimum, the surjectivity and the maximum moduli of Banach space operators. Archive classification: math.FA math.SP Mathematics Subject Classification: Primary 47B49; Secondary 47B48, 46A05, 47A10 Remarks: The abstract of this paper was posted on May 2009 in the web page of The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.0283 or http://arXiv.org/abs/0910.0283 or by email in unzipped form by transmitting an empty message with subject line uget 0910.0283 or in gzipped form by using subject line get 0910.0283 to: math(a)arXiv.org.

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Abstract of a paper by Anna Kamont and Paul F. X. Mueller
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "Rearrangements with supporting trees, isomorphisms and combinatorics of coloured dyadic intervals" by Anna Kamont and Paul F. X. Mueller. Abstract: We determine a class of rearrangements that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension. We show that there exists a large subspace of $L^p$ on which a bounded rearrangement operator acts as an isomorphism. The combinatorial issues of these problems give rise to a two-person game, to be played with colored dyadic intervals. We determine winning strategies for each of the players. Archive classification: math.FA Mathematics Subject Classification: 46B25; 46E40; 91A05 The source file(s), buch.def: 1005 bytes isoplussept091.bbl: 5771 bytes isoplussept091.tex: 98057 bytes math111.def: 7238 bytes, is(are) stored in gzipped form as 0909.4926.tar.gz with size 32kb. The corresponding postcript file has gzipped size 170kb. Submitted from: pfxm(a)bayou.uni-linz.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.4926 or http://arXiv.org/abs/0909.4926 or by email in unzipped form by transmitting an empty message with subject line uget 0909.4926 or in gzipped form by using subject line get 0909.4926 to: math(a)arXiv.org.

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Abstract of a paper by Stefano Rossi
by alspach＠fourier.math.okstate.edu 16 Oct '09

16 Oct '09

This is an announcement for the paper "On a characterization of separable dual Banach spaces through determinant subspaces of attaining-norm linear forms" by Stefano Rossi. Abstract: Necessary and sufficient conditions for a separable Banach space to be(isometrically isomorphic to) a dual space will be given. Archive classification: math.FA Remarks: 7 pages The source file(s), articolodef.tex: 24589 bytes, is(are) stored in gzipped form as 0909.4980.gz with size 8kb. The corresponding postcript file has gzipped size 66kb. Submitted from: s-rossi(a)mat.uniroma1.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.4980 or http://arXiv.org/abs/0909.4980 or by email in unzipped form by transmitting an empty message with subject line uget 0909.4980 or in gzipped form by using subject line get 0909.4980 to: math(a)arXiv.org.

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Banach October 2009