- Banach - mathdept.okstate.edu

Abstract of a paper by Simon Luecking
by alspach＠fourier.math.okstate.edu 20 Jul '10

20 Jul '10

This is an announcement for the paper "Subspaces of almost Daugavet spaces" by Simon Luecking. Abstract: We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too. Archive classification: math.FA Mathematics Subject Classification: 46B04 Remarks: 5 pages Submitted from: simon.luecking(a)fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1007.2916 or http://arXiv.org/abs/1007.2916

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Abstract of a paper by Asuman Guven Aksoy and Grzegorz Lewicki
by alspach＠fourier.math.okstate.edu 20 Jul '10

20 Jul '10

This is an announcement for the paper "Minimal projections with respect to various norms" by Asuman Guven Aksoy and Grzegorz Lewicki. Abstract: We will show that a theorem of Rudin \cite{wr1}, \cite{wr}, permits us to determine minimal projections not only with respect to the operator norm but with respect to quasi-norms in operators ideals and numerical radius in many concrete cases. Archive classification: math.FA Remarks: 16 pages Submitted from: aaksoy(a)cmc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1007.2214 or http://arXiv.org/abs/1007.2214

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Abstract of a paper by Asuman Guven Aksoy and Grzegorz Lewicki
by alspach＠fourier.math.okstate.edu 20 Jul '10

20 Jul '10

This is an announcement for the paper "Best approximation in numerical radius" by Asuman Guven Aksoy and Grzegorz Lewicki. Abstract: Let $X$ be a reflexive Banach space. In this paper we give a necessary and sufficient condition for an operator $T\in \mathcal{K}(X)$ to have the best approximation in numerical radius from the convex subset $\mathcal{U} \subset \mathcal{K}(X),$ where $\mathcal{K}(X)$ denotes the set of all linear, compact operators from $X$ into $X.$ We will also present an application to minimal extensions with respect to the numerical radius. In particular some results on best approximation in norm will be generalized to the case of the numerical radius. Archive classification: math.FA Remarks: 13 pages Submitted from: aaksoy(a)cmc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1007.2205 or http://arXiv.org/abs/1007.2205

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Abstract of a paper by Maxim V. Balashov and Dusan Repovs
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Weakly convex sets and modulus of nonconvexity" by Maxim V. Balashov and Dusan Repovs. Abstract: We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set. Archive classification: math.FA math.GN Mathematics Subject Classification: 46A55, 52A01, 52A07, 54C60, 54C65 Citation: J. Math. Anal. Appl. 371:1 (2010), 113-127 Submitted from: dusan.repovs(a)guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1007.0162 or http://arXiv.org/abs/1007.0162

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Abstract of a paper by Mikhail Lifshits and Werner Linde
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Compactness properties of weighted summation operators on trees" by Mikhail Lifshits and Werner Linde. Abstract: We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$ (V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s)\,,\quad t\in T\;, $$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g.~moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees. Archive classification: math.FA Mathematics Subject Classification: Primary: 47B06, Secondary: 06A06, 05C05 Submitted from: lifts(a)mail.rcom.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.3867 or http://arXiv.org/abs/1006.3867

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Abstract of a paper by Taras Banakh and Robert Cauty
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Topological classification of closed convex sets in Frechet spaces" by Taras Banakh and Robert Cauty. Abstract: We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, this result implies that each closed convex subset of a Frechet space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals $0\le n\le\omega$, $0\le m\le 1$ and $k\ge 0$. Archive classification: math.FA math.GN math.GT Mathematics Subject Classification: 57N17, 46A04 Remarks: 8 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.3092 or http://arXiv.org/abs/1006.3092

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Abstract of a paper by Iryna Banakh and Taras Banakh
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Constructing non-compact operators into $c_0$" by Iryna Banakh and Taras Banakh. Abstract: We prove that for each dense non-compact linear operator $S:X\to Y$ between Banach spaces there is a linear operator $T:Y\to c_0$ such that the operator $TS:X\to c_0$ is not compact. This generalizes the Josefson-Nissenzweig Theorem. Archive classification: math.FA Mathematics Subject Classification: 47B07, 46B15 Remarks: 2 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.3089 or http://arXiv.org/abs/1006.3089

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Abstract of a paper by Jesus Araujo and Luis Dubarbie
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Noncompactness and noncompleteness in isometries of Lipschitz spaces" by Jesus Araujo and Luis Dubarbie. Abstract: We solve the following two questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$: \begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \end{enumerate} In particular, we prove that requirements of completeness on $X$ and $Y$ are not necessary when $E$ and $F$ are not complete, which is in sharp contrast with results known in the scalar context. We also give the special form of this kind of isometries. Archive classification: math.FA Mathematics Subject Classification: 2010: 47B33 (Primary), 46B04, 46E15, 46E40, 47B38 (Secondary) Remarks: 14 pages, no figures, \documentclass[12pt]{amsart} Submitted from: araujoj(a)unican.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.2995 or http://arXiv.org/abs/1006.2995

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Abstract of a paper by Sergey Bobkov, Mokshay Madiman, and Liyao Wang
by alspach＠fourier.math.okstate.edu 14 Jul '10

14 Jul '10

This is an announcement for the paper "Fractional generalizations of Young and Brunn-Minkowski inequalities" by Sergey Bobkov, Mokshay Madiman, and Liyao Wang. Abstract: A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described. Archive classification: math.FA cs.IT math.IT math.PR Remarks: 17 pages Submitted from: mokshay.madiman(a)yale.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.2884 or http://arXiv.org/abs/1006.2884

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Abstract of a paper by Pandelis Dodos, Jordi Lopez-Abad and Stevo Todorcevic
by alspach＠fourier.math.okstate.edu 06 Jul '10

06 Jul '10

This is an announcement for the paper "Banach spaces and Ramsey theory: some open problems" by Pandelis Dodos, Jordi Lopez-Abad and Stevo Todorcevic. Abstract: We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them. Archive classification: math.FA math.CO Remarks: 17 pages, no figures; RACSAM, to appear Submitted from: pdodos(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.2668 or http://arXiv.org/abs/1006.2668

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