Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Measurability in C(2^k) and Kunen cardinals" by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez. Abstract: A cardinal k is called a Kunen cardinal if the sigma-algebra on k x k generated by all products AxB, coincides with the power set of k x k. For any cardinal k, let C(2^k) be the Banach space of all continuous real-valued functions on the Cantor cube 2^k. We prove that k is a Kunen cardinal if and only if the Baire sigma-algebra on C(2^k) for the pointwise convergence topology coincides with the Borel sigma-algebra on C(2^k) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given. Archive classification: math.FA Mathematics Subject Classification: 28A05, 28B05 Submitted from: avileslo(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0247 or http://arXiv.org/abs/1103.0247

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Abstract of a paper by Gideon Schechtman
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Approximate Gaussian isoperimetry for k sets" by Gideon Schechtman. Abstract: Given $2\le k\le n$, the minimal $(n-1)$-dimensional Gaussian measure of the union of the boundaries of $k$ disjoint sets of equal Gaussian measure in $\R^n$ whose union is $\R^n$ is of order $\sqrt{\log k}$. A similar results holds also for partitions of the sphere $S^{n-1}$ into $k$ sets of equal Haar measure. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15, 52A40 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/1102.4102 or http://arXiv.org/abs/1102.4102

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Abstract of a paper by Ran Levy and Gideon Schechtman
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$" by Ran Levy and Gideon Schechtman. Abstract: Let $1<p\not=2<\infty$, $\epsilon>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+\epsilon)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+\epsilon)$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\epsilon)\gamma_p$ if $p>2$ and $K_p\le (1+\epsilon)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable. Archive classification: math.FA Mathematics Subject Classification: 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/1103.0047 or http://arXiv.org/abs/1103.0047

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M. Kadec
by Vladimir Kadets 07 Mar '11

07 Mar '11

I regret to inform that my father Professor Mikhail Kadets (M. Kadec) died on Monday, March 7. He was 87 years old. Vladimir Kadets.

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Abstract of a paper by Remi Gribonval and Morten Nielsen
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "The restricted isometry property meets nonlinear approximation with redundant frames" by Remi Gribonval and Morten Nielsen. Abstract: It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP with respect to nonlinear approximation in a Hilbert space with a redundant frame. The main ingredients of our approach are: a) Jackson and Bernstein inequalities, associated to the characterization of certain approximation spaces with interpolation spaces; b) a new proof that for overcomplete frames which satisfy a Bernstein inequality, these interpolation spaces are nothing but the collection of vectors admitting a representation in the dictionary with compressible coefficients; c) the proof that the RIP implies Bernstein inequalities. As a result, we obtain that in most overcomplete random Gaussian dictionaries with fixed aspect ratio, just as in any orthonormal basis, the error of best $m$-term approximation of a vector decays at a certain rate if, and only if, the vector admits a compressible expansion in the dictionary. Yet, for mildly overcomplete dictionaries with a one-dimensional kernel, we give examples where the Bernstein inequality holds, but the same inequality fails for even the smallest perturbation of the dictionary. Archive classification: math.FA Report Number: RR-7548 Remarks: This work has been submitted for possible publication. Copyright be transferred without notice, after which this version may no longer be accessible. Submitted from: remi.gribonval(a)inria.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.5324 or http://arXiv.org/abs/1102.5324

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Abstract of a paper by Greg Kuperberg
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "Norms as a function of p are linearly independent in finite dimensions" by Greg Kuperberg. Abstract: We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by analytic continuation and a theorem of Ritt. Archive classification: math.FA Remarks: 1 page, 1 figure Submitted from: greg(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.5026 or http://arXiv.org/abs/1102.5026

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Abstract of a paper by Mikhail Ostrovskii
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "On metric characterizations of some classes of Banach spaces" by Mikhail Ostrovskii. Abstract: The paper contains the following results and observations: (1) There exists a sequence of unweighted graphs $\{G_n\}_n$ with maximum degree $3$ such that a Banach space $X$ has no nontrivial cotype iff $\{G_n\}_n$ admit uniformly bilipschitz embeddings into $X$; (2) The same for Banach spaces with no nontrivial type; (3) A sequence $\{G_n\}$ characterizing Banach spaces with no nontrivial cotype in the sense described above can be chosen to be a sequence of bounded degree expanders; (4) The infinite diamond does not admit a bilipschitz embedding into Banach spaces with the Radon-Nikod\'{y}m property; (5) A new proof of the Cheeger-Kleiner result: The Laakso space does not admit a bilipschitz embedding into Banach spaces with the Radon-Nikod\'{y}m property; (6) A new proof of the Johnson-Schechtman result: uniform bilipschitz embeddability of finite diamonds into a Banach space implies its nonsuperreflexivity. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B85, Secondary: 05C12, 46B07, 46B22, 54E35 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.5082 or http://arXiv.org/abs/1102.5082

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Frank Bonsall
by Jonathan Partington 28 Feb '11

28 Feb '11

We regret to say that Professor Frank Bonsall FRS, well known for his work in functional analysis, in particular, Banach algebras, numerical ranges and operator theory, died in Harrogate, Yorkshire on Tuesday February 22nd. He was 90 years old. Jonathan Partington, University of Leeds, UK

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Abstract of a paper by Jin Xi Chen, Zi Li Chen and Guo Xing Ji
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "Order continuous extensions of positive compact operators on Banach lattices" by Jin Xi Chen, Zi Li Chen and Guo Xing Ji. Abstract: Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and $T: G\rightarrow F$ be an order continuous positive compact (resp. weakly compact) operators. We show that if $G$ is an ideal or an order dense sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly compact) positive extension to $E$ which is likewise order continuous on $E$. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on $E$. Archive classification: math.FA Remarks: 7 pages Submitted from: jinxichen(a)home.swjtu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.4912 or http://arXiv.org/abs/1102.4912

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Abstract of a paper by A. Thiago Lopes Bernardino
by alspach＠math.okstate.edu 28 Feb '11

28 Feb '11

This is an announcement for the paper "On cotype and inclusions for absolutely summing multilinear operators" by A. Thiago Lopes Bernardino. Abstract: In this note we improve previous results on inclusion theorems for absolutely summing multilinear operators. Archive classification: math.FA Remarks: 3 pages Submitted from: thiagodcea(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.4542 or http://arXiv.org/abs/1102.4542

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