Banach July 2008

banach@mathdept.okstate.edu

2 participants
15 discussions

Abstract of a paper by Alexey I. Popov and Vladimir G. Troitsky
by alspach＠fourier.math.okstate.edu 23 Jul '08

23 Jul '08

This is an announcement for the paper "A version of Lomonosov's theorem for collections of positive operators" by Alexey I. Popov and Vladimir G. Troitsky. Abstract: It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In the case of algebras of adjoint operators on a dual Banach space, V.Lomonosov extended this as follows: without having a compact operator in the algebra, |f(Tx)| is less than or equal to the essential norm of the pre-adjoint operator T_* for all T in A. In this paper, we prove a similar extension (in case of adjoint operators) of a result of R.Drnovsek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero positive x in X and f in X* such that f(Tx) is less than or equal to the essential norm of T_* for all T in C. Archive classification: math.FA math.OA Mathematics Subject Classification: 47B65; 47A15 The source file(s), lom-drnov.tex: 31715 bytes, is(are) stored in gzipped form as 0807.3327.gz with size 10kb. The corresponding postcript file has gzipped size 86kb. Submitted from: vtroitsky(a)math.ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.3327 or http://arXiv.org/abs/0807.3327 or by email in unzipped form by transmitting an empty message with subject line uget 0807.3327 or in gzipped form by using subject line get 0807.3327 to: math(a)arXiv.org.

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Abstract of a paper by T. P. Hyt\"onen, J. L. Torrea, and D. V. Yakubovich
by alspach＠fourier.math.okstate.edu 23 Jul '08

23 Jul '08

This is an announcement for the paper "The Littlewood--Paley--Rubio de Francia property of a Banach space for the case of equal intervals" by T. P. Hyt\"onen, J. L. Torrea, and D. V. Yakubovich. Abstract: Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series. Archive classification: math.FA Mathematics Subject Classification: 42Bxx; 46B20 Remarks: To appear in The Royal Society of Edinburgh Proc. A (Mathematics) The source file(s), lpr-equal_v6_arx.tex: 41797 bytes, is(are) stored in gzipped form as 0807.2981.gz with size 14kb. The corresponding postcript file has gzipped size 97kb. Submitted from: dmitry.yakubovich(a)uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2981 or http://arXiv.org/abs/0807.2981 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2981 or in gzipped form by using subject line get 0807.2981 to: math(a)arXiv.org.

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Abstract of a paper by Hermann Pfitzner
by alspach＠fourier.math.okstate.edu 23 Jul '08

23 Jul '08

This is an announcement for the paper "Boundaries for Banach spaces determine weak compactness" by Hermann Pfitzner. Abstract: A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that boundary. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. Godefroy's Boundary Problem asks whether nevertheless both topologies have the same bounded compact sets. This paper contains the answer in the positive. Archive classification: math.FA Mathematics Subject Classification: 46B20 The source file(s), boundary.tex: 30948 bytes, is(are) stored in gzipped form as 0807.2810.gz with size 10kb. The corresponding postcript file has gzipped size 76kb. Submitted from: Hermann.Pfitzner(a)univ-orleans.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2810 or http://arXiv.org/abs/0807.2810 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2810 or in gzipped form by using subject line get 0807.2810 to: math(a)arXiv.org.

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Abstract of a paper by Anthony Weston
by alspach＠fourier.math.okstate.edu 23 Jul '08

23 Jul '08

This is an announcement for the paper "Optimal lower bounds on the maximal p-negative type of finite metric spaces" by Anthony Weston. Abstract: This article derives lower bounds on the supremal (strict) p-negative type of finite metric spaces using purely elementary techniques. The bounds depend only on the cardinality and the (scaled) diameter of the underlying finite metric space. Examples show that these lower bounds can easily be best possible under clearly delineated circumstances. We further point out that the entire theory holds (more generally) for finite semi-metric spaces without modification and wherein the lower bounds are always optimal. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B20 Remarks: 10 pages The source file(s), Gap.tex: 36066 bytes, is(are) stored in gzipped form as 0807.2705.gz with size 11kb. The corresponding postcript file has gzipped size 95kb. Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2705 or http://arXiv.org/abs/0807.2705 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2705 or in gzipped form by using subject line get 0807.2705 to: math(a)arXiv.org.

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Abstract of a paper by Spiros A. Argyros, Alexander D. Arvanitakis, and Andreas G. Tolias
by alspach＠fourier.math.okstate.edu 17 Jul '08

17 Jul '08

This is an announcement for the paper "Saturated extensions, the attractors method and Hereditarily James Tree Space" by Spiros A. Argyros, Alexander D. Arvanitakis, and Andreas G. Tolias. Abstract: In the present work we provide a variety of examples of HI Banach spaces containing no reflexive subspace and we study the structure of their duals as well as the spaces of their linear bounded operators. Our approach is based on saturated extensions of ground sets and the method of attractors. Archive classification: math.FA The source file(s), Aat6.tex: 290045 bytes, is(are) stored in gzipped form as 0807.2392.gz with size 77kb. The corresponding postcript file has gzipped size 377kb. Submitted from: sargyros(a)math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2392 or http://arXiv.org/abs/0807.2392 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2392 or in gzipped form by using subject line get 0807.2392 to: math(a)arXiv.org.

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Abstract of a paper by Spiros A. Argyros, Irene Deliyanni, and Andreas G. Tolias
by alspach＠fourier.math.okstate.edu 17 Jul '08

17 Jul '08

This is an announcement for the paper "Strictly singular non-compact diagonal operators on HI spaces" by Spiros A. Argyros, Irene Deliyanni, and Andreas G. Tolias. Abstract: We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$. \end{abstract} Archive classification: math.FA Mathematics Subject Classification: 46B28, 46B20, 46B03 The source file(s), diagonal_adt_1.tex: 147103 bytes, is(are) stored in gzipped form as 0807.2388.gz with size 39kb. The corresponding postcript file has gzipped size 213kb. Submitted from: sargyros(a)math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2388 or http://arXiv.org/abs/0807.2388 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2388 or in gzipped form by using subject line get 0807.2388 to: math(a)arXiv.org.

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Abstract of a paper by Christian Rosendal
by alspach＠fourier.math.okstate.edu 17 Jul '08

17 Jul '08

This is an announcement for the paper "An exact Ramsey principle for block sequences" by Christian Rosendal. Abstract: We prove an exact, i.e., formulated without $\Delta$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games. Archive classification: math.FA math.LO Mathematics Subject Classification: 46B03, 03E15 The source file(s), ExactRamseyPrinciples17submitted.tex: 37130 bytes, is(are) stored in gzipped form as 0807.2205.gz with size 11kb. The corresponding postcript file has gzipped size 82kb. Submitted from: rosendal(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.2205 or http://arXiv.org/abs/0807.2205 or by email in unzipped form by transmitting an empty message with subject line uget 0807.2205 or in gzipped form by using subject line get 0807.2205 to: math(a)arXiv.org.

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Abstract of a paper by William B. Johnson and Assaf Naor
by alspach＠fourier.math.okstate.edu 17 Jul '08

17 Jul '08

This is an announcement for the paper "The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite" by William B. Johnson and Assaf Naor. Abstract: Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-dimensional subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-dimensional subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{\Omega(\alpha(n))}$, where $\alpha$ is the inverse Ackermann function. Archive classification: math.FA math.MG The source file(s), JL-L3.1.TEX: 43297 bytes, is(are) stored in gzipped form as 0807.1919.gz with size 14kb. The corresponding postcript file has gzipped size 74kb. 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/0807.1919 or http://arXiv.org/abs/0807.1919 or by email in unzipped form by transmitting an empty message with subject line uget 0807.1919 or in gzipped form by using subject line get 0807.1919 to: math(a)arXiv.org.

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Abstract of a paper by Marisa Zymonopoulou
by alspach＠fourier.math.okstate.edu 10 Jul '08

10 Jul '08

This is an announcement for the paper "The complex Busemann-Petty problem for arbitrary measures" by Marisa Zymonopoulou. Abstract: The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this article we show that the answer remains the same if the volume is replaced by an "almost" arbitrary measure. This result is the complex analogue of Zvavitch's generalization to arbitrary measures of the original real Busemann-Petty problem. Archive classification: math.FA The source file(s), CBPGM.tex: 37275 bytes, is(are) stored in gzipped form as 0807.0779.gz with size 10kb. The corresponding postcript file has gzipped size 89kb. Submitted from: marisa(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.0779 or http://arXiv.org/abs/0807.0779 or by email in unzipped form by transmitting an empty message with subject line uget 0807.0779 or in gzipped form by using subject line get 0807.0779 to: math(a)arXiv.org.

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Abstract of a paper by Marisa Zymonopoulou
by alspach＠fourier.math.okstate.edu 10 Jul '08

10 Jul '08

This is an announcement for the paper "The modified complex Busemann-Petty problem on sections of convex bodies" by Marisa Zymonopoulou. Abstract: Since the answer to the complex Busemann-Petty problem is negative in most dimensions, it is natural to ask what conditions on the (n-1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions. Archive classification: math.FA The source file(s), MCBP.tex: 44421 bytes, is(are) stored in gzipped form as 0807.0776.gz with size 12kb. The corresponding postcript file has gzipped size 104kb. Submitted from: marisa(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0807.0776 or http://arXiv.org/abs/0807.0776 or by email in unzipped form by transmitting an empty message with subject line uget 0807.0776 or in gzipped form by using subject line get 0807.0776 to: math(a)arXiv.org.

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Banach July 2008