Banach February 2011

banach@mathdept.okstate.edu

2 participants
28 discussions

Abstract of a paper by Guillaume Aubrun and Ion Nechita
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "The multiplicative property characterizes $\ell_p$ and $L_p$ norms" by Guillaume Aubrun and Ion Nechita. Abstract: We show that $\ell_p$ norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the $L_p$ norms are the unique rearrangement-invariant norms on a probability space such that $\|X Y\|=\|X\|\cdot\|Y\|$ for every pair $X,Y$ of independent random variables. Our proof relies on Cram\'er's large deviation theorem. Archive classification: math.FA Remarks: 8 pages, 1 figure Submitted from: inechita(a)uottawa.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.2618 or http://arXiv.org/abs/1102.2618

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Abstract of a paper by Daniel Fresen
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "Comments on the floating body and the hyperplane conjecture" by Daniel Fresen. Abstract: We provide upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\ and the convex floating body $K_{\delta }$ inside $K$. We also discuss the hyperplane conjecture (the slicing problem) and provide a reformulation of this famous unsolved mystery in terms of the floating body. Archive classification: math.FA math.PR Mathematics Subject Classification: 52A23, 52A20, 52A21, 52A38 Remarks: 8 pages Submitted from: djfb6b(a)mail.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.2570 or http://arXiv.org/abs/1102.2570

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Abstract of a paper by Daniel Pellegrino and Joilson Ribeiro
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "On everywhere almost summing multilinear operators" by Daniel Pellegrino and Joilson Ribeiro. Abstract: In this paper we obtain new results and characterizations for the classes (ideals) of everywhere almost summing multilinear operators and everywhere almost summing $n$-homogeneous polynomials. Among other results we prove that the ideal of everywhere almost summing polynomials is a global holomorphy type (this is not true for the original concept of almost summing polynomials). 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/1102.1460 or http://arXiv.org/abs/1102.1460

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Abstract of a paper by Ralf Beckmann and Anton Deitmar
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "Strong vector valued integrals" by Ralf Beckmann and Anton Deitmar. Abstract: Strong Bochner type integrals with values in locally convex spaces are introduced. It is shown that the strong integral exists in the same cases as the weak (Gelfand-Pettis) integral is known to exist. The strong integral has better continuity properties that the weak integral. Archive classification: math.FA Submitted from: deitmar(a)uni-tuebingen.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.1246 or http://arXiv.org/abs/1102.1246

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Abstract of a paper by Richard J. Smith
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "Tree duplicates, $G_\delta$-diagonals and Gruenhage spaces" by Richard J. Smith. Abstract: We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function. Archive classification: math.FA math.GN Submitted from: richard.smith(a)ucd.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.0982 or http://arXiv.org/abs/1102.0982

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Abstract of a paper by Roman Vershynin
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Invertibility of symmetric random matrices" by Roman Vershynin. Abstract: Let H be an n by n symmetric random matrix whose above-diagonal entries are general iid random variables (possibly discrete) with zero mean, unit variance, and subgaussian tail decay. We prove that H is singular with probability at most exp(n^{-c}) for some constant c>0, and that the spectral norm of the inverse of H is O(\sqrt{n}) with high probability. More generally, the spectrum of H is delocalized -- with high probability, there are no eigenvalues in an arbitrary fixed interval of the optimal length o(n^{-1/2}). The delocalization result also holds under the fourth moment assumption on the entries of H. These results improve upon the polynomial singularity bound O(n^{-1/8+epsilon}) due to Costello, Tao and Vu, and they generalize, up to constant factors, previous results for distributions whose first few moments match the moments of the normal distribution (due to the universality results of Tao and Vu) and for continuous distributions in the bulk of the spectrum (due to Erd\"os, Schlein and Yau). Archive classification: math.PR math.FA Mathematics Subject Classification: 15B52 Remarks: 52 pages Submitted from: romanv(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.0300 or http://arXiv.org/abs/1102.0300

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Abstract of a paper by Detelin Dosev, William B. Johnson, and Gideon Schechtman
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Commutators on $L_p$, $1\le p<\infty$" by Detelin Dosev, William B. Johnson, and Gideon Schechtman. Abstract: The operators on $\LP=L_p[0,1]$, $1\leq p<\infty$, which are not commutators are those of the form $\lambda I + S$ where $\lambda\neq 0$ and $S$ belongs to the largest ideal in $\opLP$. The proof involves new structural results for operators on $\LP$ which are of independent interest. Archive classification: math.FA Mathematics Subject Classification: 47B47, 46E30 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.0137 or http://arXiv.org/abs/1102.0137

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Abstract of a paper by Matthew Daws, Richard Haydon, Thomas Schlumprecht, and Stuart White
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Shift invariant preduals of $\ell_1(\Z)$" by Matthew Daws, Richard Haydon, Thomas Schlumprecht, and Stuart White. Abstract: The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is equivalent to making the natural convolution multiplication on $\ell_1(\Z)$ separately weak$*$-continuous and so turning $\ell_1(\Z)$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_0(K)$ (for countable locally compact $K$) and $\ell_1(\Z)$ is $c_0(\Z)$. We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak$^*$-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to $c_0$. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of $\Z$. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to $c_0$. Archive classification: math.FA Remarks: 31 pages Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5696 or http://arXiv.org/abs/1101.5696

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Abstract of a paper by Maxim V. Balashov and Dusan Repovs
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order" by Maxim V. Balashov and Dusan Repovs. Abstract: We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius. Archive classification: math.FA math.GN Mathematics Subject Classification: 46C05, 54C60, 46N10, 32F17 Citation: J. Math. Anal. Appl. 377:2 (2011), 754-761 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/1101.5685 or http://arXiv.org/abs/1101.5685

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Abstract of a paper by Cleon S. Barroso, Ondrej F.K. Kalenda and Pei-Kee Lin
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "On the approximate fixed point property in abstract spaces" by Cleon S. Barroso, Ondrej F.K. Kalenda and Pei-Kee Lin. Abstract: Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual and $Z$ a subset of $X^*$. In this paper, we establish some results concerning the $\sigma(X,Z)$-approximate fixed point property for bounded, closed convex subsets $C$ of $X$. Three major situations are studied. First when $Z$ is separable in the strong topology. Second when $X$ is a metrizable locally convex space and $Z=X^*$, and third when $X$ is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fr\'echet-Urysohn property for certain sets with regarding the $\sigma(X,Z)$-topology. The support tools include the Brouwer's fixed point theorem and an analogous version of the classical Rosenthal's $\ell_1$-theorem for $\ell_1$-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces. Archive classification: math.FA math.GN Mathematics Subject Classification: 47H10, 46A03 Remarks: 14 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/1101.5274 or http://arXiv.org/abs/1101.5274

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Banach February 2011