Banach April 2010

banach@mathdept.okstate.edu

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22 discussions

Abstract of a paper by Philip A. H. Brooker
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Direct sums and the Szlenk index" by Philip A. H. Brooker. Abstract: For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $\omega^\alpha$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $\varepsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums. Archive classification: math.FA Submitted from: philip.brooker(a)anu.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5708 or http://arXiv.org/abs/1003.5708

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Abstract of a paper by Philip A. H. Brooker
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Operator ideals associated with the Szlenk index" by Philip A. H. Brooker. Abstract: For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$ consisting of all operators whose Szlenk index is an ordinal not exceeding $\omega^\alpha$. Our main result is that $\mathscr{SZ}_\alpha$ is a closed, injective, surjective operator ideal for each $\alpha$. We also study the relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known closed operator ideals. Archive classification: math.FA Submitted from: philip.brooker(a)anu.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5706 or http://arXiv.org/abs/1003.5706

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Abstract of a paper by Constantinos Kardaras
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Strictly positive support points of convex sets in $\mathbb{L}^0_+$" by Constantinos Kardaras. Abstract: We introduce the concept of strictly positive support points of convex sets in $\mathbb{L}^0_+$, the nonnegative orthant of the topological vector space $\mathbb{L}^0$ of all random variables built over a probability space. Traditional functional-analytic definitions fail, due to the fact that the topological dual of $\mathbb{L}^0$ is trivial when the underlying probability space is nonatomic. A necessary and sufficient condition for an element of a convex set in $\mathbb{L}^0_+$ to be a strictly positive support point of the set is given, inspired from ideas in financial mathematics. Archive classification: math.FA math.PR Remarks: 8 pages Submitted from: langostas(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5419 or http://arXiv.org/abs/1003.5419

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Weak$^*$ closures and derived sets in dual Banach spaces" by Mikhail I. Ostrovskii. Abstract: The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\overline{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^*$ be an absolutely convex subset. Then $A^{(1)}={\overline{A}\,}^*$. Archive classification: math.FA Mathematics Subject Classification: primary 46B10; secondary 46B15; 46B20 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5176 or http://arXiv.org/abs/1003.5176

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Abstract of a paper by Hermann Pfitzner
by alspach＠fourier.math.okstate.edu 08 Apr '10

08 Apr '10

This is an announcement for the paper "Phillips' Lemma for L-embedded Banach spaces" by Hermann Pfitzner. Abstract: In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive - is weak-weakly sequentially continuous. Archive classification: math.FA Remarks: accepted by Archiv der Mathematik, The original publication will be available at www.springerlink.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5088 or http://arXiv.org/abs/1003.5088

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Abstract of a paper by Guillaume Aubrun, Stanislaw Szarek and Elisabeth Werner
by alspach＠fourier.math.okstate.edu 08 Apr '10

08 Apr '10

This is an announcement for the paper "Hastings' additivity counterexample via Dvoretzky's theorem" by Guillaume Aubrun, Stanislaw Szarek and Elisabeth Werner. Abstract: The goal of this note is to show that Hastings' counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky's theorem on almost spherical sections of convex bodies. Archive classification: quant-ph math.FA Remarks: 11 pages Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.4925 or http://arXiv.org/abs/1003.4925

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Abstract of a paper by Tetiana V. Bosenko
by alspach＠fourier.math.okstate.edu 08 Apr '10

08 Apr '10

This is an announcement for the paper "Daugavet centers and direct sums of Banach spaces" by Tetiana V. Bosenko. Abstract: A linear continuous nonzero operator G:X->Y is a Daugavet center if every rank-1 operator T:X->Y satisfies ||G+T||=||G||+||T||. We study the case when either X or Y is a sum $X_1 \oplus_F X_2$ of two Banach spaces $X_1$ and $X_2$ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces $X_1$ and $X_2$ there exists a Daugavet center acting from $X_1\oplus_F X_2$, and the class of those F such that for some pair of spaces $X_1$ and $X_2$ there is a Daugavet center acting into $X_1\oplus_F X_2$. We also present several examples of such Daugavet centers. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04; secondary 46B20, 46B40 Remarks: 13 pages Submitted from: t.bosenko(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.4857 or http://arXiv.org/abs/1003.4857

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Abstract of a paper by Assaf Naor
by alspach＠fourier.math.okstate.edu 08 Apr '10

08 Apr '10

This is an announcement for the paper "L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry" by Assaf Naor. Abstract: We survey connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest Cut Problem is a striking example of the deep interplay between analysis, geometry, and probability on the one hand, and computational issues in discrete mathematics on the other. We explain how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. As an important illustrative example, we shall examine recently established connections to the the structure of the Heisenberg group, and the incompatibility of its Carnot-Carath\'eodory geometry with the geometry of the Lebesgue space $L_1$. Archive classification: math.MG cs.DS math.FA Remarks: To appear in Proceedings of the International Congress of Mathematicians, Hyderabad India, 2010 Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.4261 or http://arXiv.org/abs/1003.4261

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Abstract of a paper by Miguel Martin, Javier Meri, Mikhail Popov, and Beata Randrianantoanina
by alspach＠fourier.math.okstate.edu 08 Apr '10

08 Apr '10

This is an announcement for the paper "Numerical index of absolute sums of Banach spaces" by Miguel Martin, Javier Meri, Mikhail Popov, and Beata Randrianantoanina. Abstract: We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of $c_0$-, $\ell_1$- and $\ell_\infty$-sums and giving as a new result the case of $E$-sums where $E$ has the RNP and $n(E)=1$ (in particular for finite-dimensional $E$ with $n(E)=1$). We also show that the numerical index of a Banach space $Z$ which contains a dense increasing union of one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that the numerical indices of all infinite-dimensional $L_p(\mu)$-spaces coincide. Archive classification: math.FA Remarks: 19 pages Submitted from: randrib(a)muohio.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.3269 or http://arXiv.org/abs/1003.3269

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Abstract of a paper by Mark Rudelson and Roman Vershynin
by alspach＠fourier.math.okstate.edu 02 Apr '10

02 Apr '10

This is an announcement for the paper "Non-asymptotic theory of random matrices: extreme singular values" by Mark Rudelson and Roman Vershynin. Abstract: The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value). Archive classification: math.FA Mathematics Subject Classification: 46B09; 60B20 Remarks: Submission for International Congress of Mathematicians, Hydebabad, India, 2010 Submitted from: romanv(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.2990 or http://arXiv.org/abs/1003.2990

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Banach April 2010