- Banach - mathdept.okstate.edu

Abstract of a paper by Thomas Schlumprecht
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "On the closed subideals of $L(\ell_p\oplus\ell_q)$" by Thomas Schlumprecht. Abstract: In this paper we first review the known results about the closed subideals of the space of bounded operator on $\ell_p\oplus \ell_q$, $1<p<q<\infty$, and then construct several new ones. Archive classification: math.FA Mathematics Subject Classification: Primary: 47L20. Secondary: 47B10, 47B37 Remarks: To appear in Operators and Matrices Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.3610 or http://arXiv.org/abs/1105.3610

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Abstract of a paper by Pavel Ludvik and Jiri Spurny
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Descriptive properties of elements of biduals of Banach spaces" by Pavel Ludvik and Jiri Spurny. Abstract: If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of $x^{**}$ are quite often determined by the behaviour of $x^{**}$ on the set of extreme points of $B_{E^*}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove several results on relation between Baire classes and intrinsic Baire classes of $L_1$-preduals which were introduced by S.A. Argyros, G. Godefroy and H.P. Rosenthal. Also, several examples witnessing natural limits of our positive results are presented. Archive classification: math.FA Mathematics Subject Classification: 46B99, 46A55, 26A21 Submitted from: spurny(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.3413 or http://arXiv.org/abs/1105.3413

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Abstract of a paper by Qinggang Ren
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Coarse embedding into uniformly convex Banach space" by Qinggang Ren. Abstract: In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly relative hyperbolic to a subgroup $H$, we proved that if $H$ admits a coarse embedding into a uniformly convex Banach space, so is $B(n)=\{g\in G|\abs{g}_{S\cup\mathscr{H}}\leq n\}$. Archive classification: math.MG math.FA Remarks: 14 pages Submitted from: qinggang.ren(a)hw4.ecs.kyoto-u.ac.jp The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.3263 or http://arXiv.org/abs/1105.3263

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Abstract of a paper by Oleg Reinov and Qaisar Latif
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Grothendieck-Lidskii theorem for subspaces and factor spaces of L_p-spaces" by Oleg Reinov and Qaisar Latif. Abstract: In 1955, A. Grothendieck has shown that if the linear operator $T$ in a Banach subspace of an $L_\infty$-space is $2/3$-nuclear then the trace of $T$ is well defined and is equal to the sum of all eigenvalues $\{\mu_k(T)\}$ of $T.$ V.B. Lidski\v{\i} , in 1959, proved his famous theorem on the coincidence of the trace of the $S_1$-operator in $L_2(\nu)$ with its spectral trace $\sum_{k=1}^\infty \mu_k(T).$ We show that for $p\in[1,\infty]$ and $s\in (0,1]$ with $1/s=1+|1/2-1/p|,$ and for every $s$-nuclear operator $T$ in every subspace of any $L_p(\nu)$-space the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$ Archive classification: math.FA Mathematics Subject Classification: 47B06 Remarks: LaTeX2e, 5 pages Submitted from: orein51(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2914 or http://arXiv.org/abs/1105.2914

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Abstract of a paper by Justin Jenkinson and Elisabeth Werner
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Relative entropies for convex bodies" by Justin Jenkinson and Elisabeth Werner. Abstract: We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities. Such interpretations were first given by Paouris and Werner for symmetric convex bodies in the context of the $L_p$-centroid bodies. There, the relative entropies appear after performing second order expansions of certain expressions. Now, no symmetry assumptions are needed. Moreover, using the new bodies, already first order expansions make the relative entropies appear. Thus, these bodies detect ``faster" details of the boundary of a convex body than the $L_p$-centroid bodies. Archive classification: math.FA Submitted from: elisabeth.werner(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2846 or http://arXiv.org/abs/1105.2846

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Abstract of a paper by Cleon Barroso, Geraldo Botelho, Vinicius V. Favaro and Daniel Pellegrino
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Spaceability for the weak form of Peano's theorem and vector-valued sequence spaces" by Cleon Barroso, Geraldo Botelho, Vinicius V. Favaro and Daniel Pellegrino. Abstract: Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano's theorem on the existence of solutions of the ODE $u'=f(u)$ on $c_0$. The space of all continuous vector fields $f$ on $c_0$ is proved to contain a closed $\bf c$-dimensional subspace formed by fields $f$ for which -- except for the null field -- the weak form of Peano's theorem fails to be true. The second application generalizes known results on the existence of closed $\bf c$-dimensional subspaces inside certain subsets of $\ell_p(X)$-spaces, $0 < p < \infty$, to the existence of closed subspaces of maximal dimension inside such subsets. Archive classification: math.FA Remarks: 10 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2845 or http://arXiv.org/abs/1105.2845

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Abstract of a paper by S.A. Argyros, V. Kanellopoulos and K. Tyros
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Finite order spreading models" by S.A. Argyros, V. Kanellopoulos and K. Tyros. Abstract: Extending the classical notion of the spreading model, the $k$-spreading models of a Banach space are introduced, for every $k\in\mathbb{N}$. The definition, which is based on the $k$-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of $c_0$ and the composition property are established. As consequence, a problem concerning the structure of the $k$-iterated spreading models is solved. Archive classification: math.FA Remarks: 41 pages, no figures Submitted from: chcost(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2732 or http://arXiv.org/abs/1105.2732

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Abstract of a paper by Spiros A. Argyros and Pavlos Motakis
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Examples of k-iterated spreading models" by Spiros A. Argyros and Pavlos Motakis. Abstract: It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one. Archive classification: math.FA Remarks: 16 pages, no figures Submitted from: pmotakis(a)central.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2714 or http://arXiv.org/abs/1105.2714

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Abstract of a paper by A. Koldobsky, G. Paouris and M. Zymonopoulou
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Isomorphic properties of intersection bodies" by A. Koldobsky, G. Paouris and M. Zymonopoulou. Abstract: We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies. Archive classification: math.FA Submitted from: marisa.zym(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2629 or http://arXiv.org/abs/1105.2629

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Abstract of a paper by Jose L. Gamez-Merino, Gustavo A. Munoz-Fernandez, Daniel Pellegrino and Juan B. Seoane-Sepulveda
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Bounded and unbounded polynomials and multilinear forms: Characterizing continuity" by Jose L. Gamez-Merino, Gustavo A. Munoz-Fernandez, Daniel Pellegrino and Juan B. Seoane-Sepulveda. Abstract: In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered. Archive classification: math.FA Remarks: 8 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.1737 or http://arXiv.org/abs/1105.1737

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