Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez
by alspach＠math.okstate.edu 30 Jun '14

30 Jun '14

This is an announcement for the paper "Tukey classification of some ideals in $\omega$ and the lattices of weakly compact sets in Banach spaces" by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez. Abstract: We study the lattice structure of the family of weakly compact subsets of the unit ball $B_X$ of a separable Banach space $X$, equipped with the inclusion relation (this structure is denoted by $\mathcal{K}(B_X)$) and also with the parametrized family of almost inclusion relations $K \subseteq L+\epsilon B_X$, where $\epsilon>0$ (this structure is denoted by $\mathcal{AK}(B_X)$). Tukey equivalence between partially ordered sets and a suitable extension to deal with $\mathcal{AK}(B_X)$ are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: $\mathcal{K}(B_X)$ is equivalent either to a singleton, or to $\omega^\omega$, or to the family $\mathcal{K}(\mathbb{Q})$ of compact subsets of the rational numbers, or to the family $[\mathfrak{c}]^{<\omega}$ of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of $\mathcal{AK}(B_X)$ is obtained. For separable Banach spaces not containing $\ell^1$, we prove in ZFC that $\mathcal{K}(B_X) \sim \mathcal{AK}(B_X)$ are equivalent to either $\{0\}$, $\omega^\omega$, $\mathcal{K}(\mathbb{Q})$ or $[\mathfrak{c}]^{<\omega}$. The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied. Archive classification: math.FA math.LO Mathematics Subject Classification: 46B20(Primary), 03E60, 03E75, 06A06, 46B50, 03E75 Submitted from: avileslo(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.5526 or http://arXiv.org/abs/1406.5526

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Abstract of a paper by Khalil Saadi
by alspach＠math.okstate.edu 30 Jun '14

30 Jun '14

This is an announcement for the paper "Some properties for Lipschitz strongly p-summing operators" by Khalil Saadi. Abstract: We consider the space of molecules endowed with the transpose version of the Chevet-Saphar norm and we identify its dual space with the space of Lipschitz strongly p-summing operators. We also extend some old results to the category of Lipschitz mappings and we give a factorization result of Lipschitz (p,r,s)-summing operators. Archive classification: math.FA Mathematics Subject Classification: [2000] 47B10, 46B28, 47L20 Remarks: 19 pages Submitted from: kh_saadi(a)yahoo.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.5551 or http://arXiv.org/abs/1406.5551

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Abstract of a paper by S. Astashkin, F. Sukochev, and D. Zanin
by alspach＠math.okstate.edu 30 Jun '14

30 Jun '14

This is an announcement for the paper "On uniqueness of distribution of a random variable whose independent copies span a subspace in L_p" by S. Astashkin, F. Sukochev, and D. Zanin. Abstract: Let 1\leq p<2 and let L_p=L_p[0,1] be the classical L_p-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable f from L_p spans in L_p a subspace isomorphic to some Orlicz sequence space l_M. We present precise connections between M and f and establish conditions under which the distribution of a random variable f whose independent copies span l_M in L_p is essentially unique. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B09 Remarks: 14 pages, submitted Submitted from: astash(a)samsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.4950 or http://arXiv.org/abs/1406.4950

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Abstract of a paper by Michael Cwikel, Mario Milman and Richard Rochberg
by alspach＠math.okstate.edu 30 Jun '14

30 Jun '14

This is an announcement for the paper "A brief survey of Nigel Kalton's work on interpolation and related topics" by Michael Cwikel, Mario Milman and Richard Rochberg. Abstract: This is the third of a series of papers surveying some small part of the remarkable work of our friend and colleague Nigel Kalton. We have written it as part of a tribute to his memory. It does not contain new results. This time, rather than concentrating on one particular paper, we attempt to give a general overview of Nigel's many contributions to the theory of interpolation of Banach spaces, and also, significantly, quasi-Banach spaces. Archive classification: math.FA Mathematics Subject Classification: Primary 46B70, 46A16. Secondary 42B20, 42B30 Remarks: 11 pages Submitted from: mcwikel(a)math.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.3842 or http://arXiv.org/abs/1406.3842

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Abstract of a paper by Amir Bahman Nasseri
by alspach＠math.okstate.edu 30 Jun '14

30 Jun '14

This is an announcement for the paper "The spectrum of operators on C(K) with the Grothendieck Property and characterization of J-class operators which are adjoints" by Amir Bahman Nasseri. Abstract: This article deals with properties of spectra of operators on C(K)-spaces with the Grothendieck property (e.g. l^{\infty}) and application to so called J-class operators introduced by A. Manoussos and G. Costakis. We will show that C(K) has the Grothendieck property if and only if the boundary of the spectrum of every operator on C(K) consists entirely of eigenvalues of its adjoint. As a consequence we will see that there does not exist invertible J-class operators on C(K) with the Grothendieck property. In the third section we will give a quantitative and qualitative characterization of all J-class operators on l^{\infty} which are adjoints from operators on l^1. Archive classification: math.SP math.DS math.FA Remarks: 19 pages Submitted from: nasseri(a)uni-wuppertal.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.3815 or http://arXiv.org/abs/1406.3815

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School and conference 2nd announcement: Besancon, Autumn 2014
by Tony Prochazka 26 Jun '14

26 Jun '14

Dear colleagues, This is the second annoncement of the two following closely related events. 1) The *Autum school on "Nonlinear geometry of Banach spaces and applications"*, in Metabief, France (October 20-24, 2014). The following mathematicians have kindly accepted our invitation to deliver a short course: Gilles Godefroy (Université Paris 6), Petr Hajek (Czech Academy of Sciences and Czech Technical University), Mikhail Ostrovskii (St. John's University, New York), Nirina Lovasoa Randrianarivony (Saint Louis University - to be confirmed), Guoliang Yu (Texas A&M University). Web: http://trimestres-lmb.univ-fcomte.fr/Autumn-School-on-Nonlinear.html?lang=en Registration: Open until September 5. 2) The *conference on "Geometric functional analysis and its applications"* in Besancon, France (October 27-31, 2014). The following main speakers have already agreed to deliver a plenary lecture: Fernando Albiac (Univ. Publica de Navarra), Florent Baudier (Texas A&M University, Paris 6) , Robert Deville (Univ. Bordeaux) , Stephen Dilworth (Univ. South Carolina), Valentin Ferenczi (Univ. Sao Paulo) , Bill Johnson (Texas A&M University), Beata Randrianantoanina (Miami Univ Ohio), Gideon Schechtman (Weizmann Institute), Thomas Schlumprecht (Texas A&M University), Alain Valette (Univ. Neuchatel). Web: http://trimestres-lmb.univ-fcomte.fr/Autumn-School-on-Nonlinear.html?lang=en Registration: Open until September 30. Participants will have the opportunity to give a short talk. The deadline for abstract submission is September 20. The purpose of these meetings is to bring together researchers and students with common interest in the field. They will offer many possibilities for informal discussions. Graduate students and others beginning their mathematical career are encouraged to participate. Thes two events are part of the trimester on "Geometric and noncommutative methods in functional analysis" organized by the "Laboratoire de Mathematiques de Besancon" during the Autumn 2014, see http://trimestres-lmb.univ-fcomte.fr/af.html . We are looking forward to meeting you! The organizers, Gilles Lancien and Tony Prochazka

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 16 Jun '14

16 Jun '14

This is an announcement for the paper "Connections between metric characterizations of superreflexivity and Radon-Nikod\'ym property for dual Banach spaces" by Mikhail I. Ostrovskii. Abstract: Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikod\'ym property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B85 (primary), 46B07, 46B22 (secondary) Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.0904 or http://arXiv.org/abs/1406.0904

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Abstract of a paper by Hana Bendova, Ondrej F.K. Kalenda and Jiri Spurny
by alspach＠math.okstate.edu 16 Jun '14

16 Jun '14

This is an announcement for the paper "Quantification of the Banach-Saks property" by Hana Bendova, Ondrej F.K. Kalenda and Jiri Spurny. Abstract: We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for $\ell_1$-spreading models. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 16 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.0684 or http://arXiv.org/abs/1406.0684

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Abstract of a paper by Herve Queffelec, and Kristian Seip
by alspach＠math.okstate.edu 16 Jun '14

16 Jun '14

This is an announcement for the paper "Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series" by Herve Queffelec, and Kristian Seip. Abstract: By a theorem of Bayart, $\varphi$ generates a bounded composition operator on the Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if $\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series with the following mapping properties: $\psi$ maps the right half-plane into the half-plane $\Real s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\Hp$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory ($s$-numbers, type and ecotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for $\Ht$, developed in an earlier paper, using estimates of solutions of the $\overline{\partial}$ equation. A transference principle from $H^p$ of the unit disc is discussed, leading to explicit examples of compact composition operators on $\Ho$ with approximation numbers decaying at a variety of sub-exponential rates. Archive classification: math.FA math.CV Submitted from: seip(a)math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1406.0445 or http://arXiv.org/abs/1406.0445

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Abstract of a paper by Maria D. Acosta
by alspach＠math.okstate.edu 16 Jun '14

16 Jun '14

This is an announcement for the paper "The Bishop-Phelps-Bollob\'{a}s property for operators on $C(K)$" by Maria D. Acosta. Abstract: We provide a version for operators of the Bishop-Phelps-Bollob\'{a}s Theorem when the domain space is the complex space $C_0(L)$. In fact we prove that the space of weakly compact operators from the complex space $C_0(L)$ into a ${\mathbb C}$-uniformly convex space satisfies the Bishop-Phelps-Bollob\'{a}s property for operators. As a consequence, in the complex case, the space of operators from $C_0(L)$ into $L_p (\mu)$ ($1 \le p < \infty $) satisfies the Bishop-Phelps-Bollob\'{a}s property for operators. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B28, 47B99 Submitted from: dacosta(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1405.6428 or http://arXiv.org/abs/1405.6428

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