February 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by Grzegorz Plebanek
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On isomorphisms of Banach spaces of continuous functions" by Grzegorz Plebanek. Abstract: We prove that if $K$ and $L$ are compact spaces and $C(K)$ and $C(L)$ are isomorphic as Banach spaces then $K$ has a $\pi$-base consisting of open sets $U$ such that $\overline{U}$ is a continuous image of some compact subspace of $L$. This gives some information on isomorphic classes of the spaces of the form $C([0,1]^\kappa)$ and $C(K)$ where $K$ is Corson compact. Archive classification: math.FA Mathematics Subject Classification: Primary 46B26, 46B03, 46E15 Remarks: 15 pages Submitted from: grzes(a)math.uni.wroc.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.3211 or http://arXiv.org/abs/1302.3211

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Abstract of a paper by Miek Messerschmidt and Marcel de Jeu
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Right inverses of surjections from cones onto Banach spaces" by Miek Messerschmidt and Marcel de Jeu. Abstract: Abstract. We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; an improved version of the usual Open Mapping Theorem is then a special case. As another consequence, a stronger version of the analogue of And\^o's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in And\^o's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, Secondary 46A30, 46B20, 46B40 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2822 or http://arXiv.org/abs/1302.2822

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Abstract of a paper by Hu Bingyang, Le Hai Khoi, and Kehe Zhu
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Frames and operators in Schatten classes" by Hu Bingyang, Le Hai Khoi, and Kehe Zhu. Abstract: Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\|Tf_n\|\}\in \ell^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0<p\le2$, $T$ belongs to $S_p$ if and only if $\{\|Tf_n\|\}\in\ell^p$ for \emph{some} frame $\{f_n\}$ in $H$. Similar conditions are also obtained in terms of the sequence $\{\langle Tf_n,f_n\rangle\}$ and the double-indexed sequence $\{\langle Tf_n,f_m\rangle\}$. Archive classification: math.FA Mathematics Subject Classification: 47B10, 46A35, 46B15 Remarks: 27 pages Submitted from: kzhu(a)math.albany.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2490 or http://arXiv.org/abs/1302.2490

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Abstract of a paper by Stanislaw J. Szarek
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On measures of symmetry and floating bodies" by Stanislaw J. Szarek. Abstract: We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio of the volumes of the two parts is at most $\rho(K)$. It is well known that $\rho(K)=1$ iff K is symmetric. We establish a precise upper bound on $\rho(K)$; this recovers a 1960 result of Grunbaum. We also provide a characterization of equality cases (relevant to recent results of Nill and Paffenholz about toric varieties) and relate these questions to the concept of convex floating bodies. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20, 52A40, 46B20 Remarks: 5 pages; this is a slightly edited manuscript from early '00s The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2076 or http://arXiv.org/abs/1302.2076

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Abstract of a paper by Benoit Kloeckner
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Yet another short proof of Bourgain's distorsion estimate" by Benoit Kloeckner. Abstract: We use a self-improvement argument to give a very short and elementary proof of the result of Bourgain saying that regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces. Archive classification: math.FA math.MG Report Number: IFPREPUB Remarks: 2 pages. Submitted from: benoit.kloeckner(a)ens-lyon.org The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.1738 or http://arXiv.org/abs/1302.1738

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Abstract of a paper by Wieslaw Kubis, Anibal Molto, and Stanimir Troyanski
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Topological properties of the continuous function spaces on some ordered compacta" by Wieslaw Kubis, Anibal Molto, and Stanimir Troyanski. Abstract: Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter. Archive classification: math.FA math.GN Mathematics Subject Classification: 46B26, 03G10 Remarks: 11 pages 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/1302.0829 or http://arXiv.org/abs/1302.0829

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Abstract of a paper by Vitali Milman and Liran Rotem
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "alpha-concave functions and a functional extension of mixed volumes" by Vitali Milman and Liran Rotem. Abstract: Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation + on the class of quasi-concave functions, such that every class of \alpha-concave functions is closed under +. We then define the mixed integrals, which are the polarization of the integral with respect to +. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to \alpha-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form. Archive classification: math.FA math.MG Citation: Electron. Res. Announc. Math. Sci. 20 (2013), 1-11 Submitted from: liranro1(a)post.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.0823 or http://arXiv.org/abs/1302.0823

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Abstract of a paper by Spiros A. Argyros and Pavlos Motakis
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Non separable reflexive spaces admitting $\ell_1$ as a unique spreading model" by Spiros A. Argyros and Pavlos Motakis. Abstract: Examples of non separable reflexive Banach spaces $\mathfrak{X}_{2^{\aleph_0}}$, admitting only $\ell_1$ as a spreading model, are presented. The definition of the spaces is based on $\alpha$-large, $\alpha<\omega_1$ compact families of finite subsets of the continuum. We show the existence of such families and we study their properties. Moreover, based on those families we construct a reflexive space $\mathfrak{X}_{2^{\aleph_0}}^\alpha$, $\alpha<\omega_1$ with density the continuum, such that every bounded non norm convergent sequence $\{x_k\}_k$ has a subsequence generating $\ell_1^\alpha$ as a spreading model. Archive classification: math.FA math.CO Mathematics Subject Classification: 46B03, 46B06, 46B26, 03E05 Remarks: 23 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/1302.0715 or http://arXiv.org/abs/1302.0715

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Abstract of a paper by V. P. Fonf, A. J. Pallares, R. J. Smith, and S. Troyanski
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Polyhedrality in Pieces" by V. P. Fonf, A. J. Pallares, R. J. Smith, and S. Troyanski. Abstract: The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere S_X or of certain subsets of the dual ball of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K) spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the fi?nal section. Archive classification: math.FA Submitted from: apall(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.0160 or http://arXiv.org/abs/1302.0160

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Abstract of a paper by Vladimir Temlyakov
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "An inequality for the entropy numbers and its application" by Vladimir Temlyakov. Abstract: We prove an inequality for the entropy numbers in terms of nonlinear Kolmogorov's widths. This inequality is in a spirit of known inequalities of this type and it is adjusted to the form convenient in applications for $m$-term approximations with respect to a given system. Also, we obtain upper bounds for the $m$-term approximation by the Weak Relaxed Greedy Algorithm with respect to a system which is not a dictionary. Archive classification: math.MG math.FA Submitted from: n.i.pentacaput(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.7624 or http://arXiv.org/abs/1301.7624

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