Banach July 2012

banach@mathdept.okstate.edu

3 participants
26 discussions

Abstract of a paper by Richard Lechner
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "The one-third-trick and shift operators" by Richard Lechner. Abstract: In this paper we obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation. Archive classification: math.FA Submitted from: lechner(a)bayou.uni-linz.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.2347 or http://arXiv.org/abs/1207.2347

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Abstract of a paper by Anil Kumar Karn and Deba Prasad Sinha
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "Compactness and an approximation property related to an operator ideal" by Anil Kumar Karn and Deba Prasad Sinha. Abstract: For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell _{\infty}$-extension and $\ell_1$-lifting properties for an operator ideal $\mathcal A$ and study ${\mathcal A}\circ{\mathcal K}$, ${\mathcal }\circ{\mathcal A}$ and the $\mathcal A$-approximation property where $\mathcal A$ is injective or surjective and/or with the $\ell _{\infty}$-extension or $\ell _1$-lifting property. In particular, we show that if $\mathcal A$ is an injective operator ideal with the $\ell _\infty$-extension property, then we have: {\item{(a)} $X$ has the $\mathcal A$-approximation property if and only if $({\mathcal A}^{min})^{inj}(Y,X)={\mathcal A}^{min}(Y,X)$, for all Banach spaces $Y$. \item{(b)} The dual space $X^*$ has the $\mathcal A$-approximation property if and only if $(({\mathcal A}^{dual})^{min})^{sur}(X,Y)=({\mathcal A}^{dual})^{min}(X,Y)$, for all Banach spaces $Y$.}For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, Archive classification: math.FA Mathematics Subject Classification: Primary 46B50, Secondary 46B20, 46B28, 47B07 Remarks: 23 pages Submitted from: anilkarn(a)niser.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.1947 or http://arXiv.org/abs/1207.1947

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Abstract of a paper by Gilles Lancien and Eva Pernecka
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "Approximation properties and Schauder decompositions in Lipschitz-free spaces" by Gilles Lancien and Eva Pernecka. Abstract: We prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. We also show that the Lipschitz-free spaces over $\ell_1^N$ or $\ell_1$ have monotone finite-dimensional Schauder decompositions. Archive classification: math.FA Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.1583 or http://arXiv.org/abs/1207.1583

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Abstract of a paper by Gustavo Garrigos, Eugenio Hernandez, and Timur Oikhberg
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "Lebesgue type inequalities for quasi-greedy bases" by Gustavo Garrigos, Eugenio Hernandez, and Timur Oikhberg. Abstract: We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases. Archive classification: math.FA Mathematics Subject Classification: 41A65, 41A46, 41A17 Report Number: 01 Remarks: 19 pages Submitted from: eugenio.hernandez(a)uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.0946 or http://arXiv.org/abs/1207.0946

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Abstract of a paper by Daniel Pellegrino, Juan Seoane-Sepulveda and Diana M. Serrano-Rodriguez
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "There exist multilinear Bohnenblust--Hille constants $(C_{n})_{n=1}^{\infty}$ with $\displaystyle \lim_{n\rightarrow \infty}(C_{n+1}-C_{n}) =0.$" by Daniel Pellegrino, Juan Seoane-Sepulveda and Diana M. Serrano-Rodriguez. Abstract: After almost 80 decades of dormancy, the Bohnenblust--Hille inequalities have experienced an effervescence of new results and sightly applications in the last years. The multilinear version of the Bohnenblust--Hille inequality asserts that for every positive integer $m\geq1$ there exists a sequence of positive constants $C_{m}\geq1$ such that% \[ \left( \sum\limits_{i_{1},\ldots,i_{m}=1}^{N}\left\vert U(e_{i_{^{1}}}% ,\ldots,e_{i_{m}})\right\vert ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C_{m}\sup_{z_{1},\ldots,z_{m}\in\mathbb{D}^{N}}\left\vert U(z_{1},\ldots ,z_{m})\right\vert \] for all $m$-linear forms $U:\mathbb{C}^{N}\times\cdots\times\mathbb{C}% ^{N}\rightarrow\mathbb{C}$ and positive integers $N$ (the same holds with slightly different constants for real scalars). The first estimates obtained for $C_{m}$ showed exponential growth but, only very recently, a striking new panorama emerged: the polynomial Bohnenblust--Hille inequality is hypercontractive and the multilinear Bohnenblust--Hille inequality is subexponential. Despite all recent advances, the existence of a family of constants $\left( C_{m}\right) _{m=1}^{\infty}$ so that \[ \lim_{n\rightarrow\infty}\left( C_{n+1}-C_{n}\right) =0 \] has not been proved yet. The main result of this paper proves that such constants do exist. As a consequence of this, we obtain new information on the optimal constants $\left( K_{n}\right) _{n=1}^{\infty}$ satisfying the multilinear Bohnenblust--Hille inequality. Let $\gamma$ be Euler's famous constant; for any $\varepsilon>0$, we show that \[ K_{n+1}-K_{n}\leq\left( 2\sqrt{2}-4e^{\frac{1}{2}\gamma-1}\right) n^{\log_{2}\left( 2^{-3/2}e^{1-\frac{1}{2}\gamma}\right) +\varepsilon}, \] for infinitely many $n$. Numerically, choosing a small $\varepsilon$, \[ K_{n+1}-K_{n}\leq0.8646\left( \frac{1}{n}\right) ^{0.4737}% \] for infinitely many $n.$ Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.0124 or http://arXiv.org/abs/1207.0124

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Abstract of a paper by Volker Wilhelm Thurey
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "Angles and a classification of normed spaces" by Volker Wilhelm Thurey. Abstract: We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the special case of real inner product spaces. With these different angles we achieve a classification of normed spaces, and we obtain a characterization of inner product spaces. Finally we consider this construction also for a generalization of normed spaces, i.e. for spaces which may have a non-convex unit ball. Archive classification: math.FA Mathematics Subject Classification: 2010 AMS-classification: 46B20, 52A10 Remarks: 23 pages, 1 figure Submitted from: volker(a)thuerey.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.0074 or http://arXiv.org/abs/1207.0074

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Banach July 2012