Banach February 2008

banach@mathdept.okstate.edu

2 participants
23 discussions

Abstract of a paper by Gilles Pisier
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Complex interpolation between Hilbert, Banach and operator spaces" by Gilles Pisier. Abstract: Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property:\ there is a function $\vp\to \Delta_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon \ L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e.\ of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le \Delta_X(\vp)$ on $L_2(X)$. We show that $\Delta_X(\vp)\in O(\vp^\alpha)$ for some $\alpha>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $\theta$-Hilbertian spaces for some $ \theta>0$ (see Corollary \ref{comcor4.3}), where $\theta$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. Let $B_{{r}}(L_2(\mu))$ be the space of all regular operators on $L_2(\mu)$. We are able to describe the complex interpolation space \[ (B_{{r}}(L_2(\mu), B(L_2(\mu))^\theta. \] We show that $T\colon \ L_2(\mu)\to L_2(\mu)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $\theta$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\ell_{p_0}), B(\ell_{p_1}))^\theta \ {\rm or}\ (B(L_{p_0}), B(L_{p_1}))^\theta $$ for any pair $1\le p_0,p_1\le \infty$ and $0<\theta<1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp.\ $PM(G)$) be the space of complex measures (resp.\ pseudo-measures) on $G$ equipped with the usual norm $\|\mu\|_{M(G)} = |\mu|(G)$ (resp. \[ \|\mu\|_{PM(G)} = \sup\{|\hat\mu(\gamma)| \ \big| \ \gamma\in\widehat G\}). \] We describe similarly the interpolation space $(M(G), PM(G))^\theta$. Various extensions and variants of this result will be given, e.g.\ to Schur multipliers on $B(\ell_2)$ and to operator spaces. Archive classification: math.FA math.OA The source file(s), complex.4fev08.tex: 174268 bytes, is(are) stored in gzipped form as 0802.0476.gz with size 51kb. The corresponding postcript file has gzipped size 253kb. Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0802.0476 or http://arXiv.org/abs/0802.0476 or by email in unzipped form by transmitting an empty message with subject line uget 0802.0476 or in gzipped form by using subject line get 0802.0476 to: math(a)arXiv.org.

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Abstract of a paper by Olivier Guedon, Shahar Mendelson, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Majorizing measures and proportional subsets of bounded orthonormal systems" by Olivier Guedon, Shahar Mendelson, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures. Archive classification: math.FA math.PR The source file(s), arXiv.tex: 50357 bytes, is(are) stored in gzipped form as 0801.3556.gz with size 16kb. The corresponding postcript file has gzipped size 130kb. Submitted from: alain.pajor(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.3556 or http://arXiv.org/abs/0801.3556 or by email in unzipped form by transmitting an empty message with subject line uget 0801.3556 or in gzipped form by using subject line get 0801.3556 to: math(a)arXiv.org.

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Abstract of a paper by A. Ibort, P. Linares, and J. G. Llavona
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Continuous multilinear functionals on $C(K)$-spaces are integral" by A. Ibort, P. Linares, and J. G. Llavona. Abstract: In this paper we prove the theorem stated on the title: every continuous multilinear functional on $C(K)$-spaces is integral, or what is the same any polymeasure defined on the product of Borelian $\sigma$-algebras defined on compact sets can be extended to a bounded Borel measure on the compact product space. We provide two different proofs of the same result, each one stressing a different aspect of the various implications of this fact. The first one, valid for compact subsets of $\R^n$, is based on the classical multivariate theory of moments and is a natural extension of the Hausdorff moment problem to multilinear functionals. The second proof relies on a multilinear extension of the decomposition theorem of linear functionals on its positive and negative part which allows us prove a multilinear Riesz Theorem as well. These arguments are valid for arbitrary Hausdorff compact sets. Archive classification: math.FA Mathematics Subject Classification: 46G25 Remarks: 10 pages The source file(s), Integralmultilinear.tex: 39365 bytes, is(are) stored in gzipped form as 0801.2878.gz with size 13kb. The corresponding postcript file has gzipped size 85kb. Submitted from: plinares(a)mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2878 or http://arXiv.org/abs/0801.2878 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2878 or in gzipped form by using subject line get 0801.2878 to: math(a)arXiv.org.

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Abstract of a paper by O. Hirzallah, F. Kittaneh, and M. S. Moslehian
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Schatten p-norm inequalities related to a characterization of inner product spaces" by O. Hirzallah, F. Kittaneh, and M. S. Moslehian. Abstract: Let $A_1, \cdots A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, \cdots A_n$ belong to a Schatten $p$-class, for some $p>0$, then \begin{equation*} 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p \end{equation*} for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \begin{equation*} \sum_{i,j=1}^n\|A_i\pm A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p \end{equation*} for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch. Archive classification: math.OA math.FA Mathematics Subject Classification: 46C15, 47A30, 47B10, 47B15 Remarks: 6 pages The source file(s), Schattenp-norminequalitiesrelatedtoacharacteriztionofinnerproductspaces.tex: 14968 bytes, is(are) stored in gzipped form as 0801.2726.gz with size 4kb. The corresponding postcript file has gzipped size 56kb. Submitted from: moslehian(a)ferdowsi.um.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2726 or http://arXiv.org/abs/0801.2726 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2726 or in gzipped form by using subject line get 0801.2726 to: math(a)arXiv.org.

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Abstract of a paper by Jesus Araujo and Juan J. Font
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Stability and instability of weighted composition operators" by Jesus Araujo and Juan J. Font. Abstract: Let $\epsilon >0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $\epsilon$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le \epsilon$, whenever $f,g\in C(X)$ satisfy $\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$ and $fg\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $\epsilon$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $\epsilon$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($\epsilon$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases. Archive classification: math.FA Mathematics Subject Classification: Primary 47B38; Secondary 46J10, 47B33 Remarks: 37 pages, 7 figures. A beamer presentation at www.araujo.tk The source file(s), ejemploy0d.eps: 10802 bytes stability86.tex: 91977 bytes total2gabove.eps: 20323 bytes total2i.eps: 20467 bytes w01c.eps: 9921 bytes w11d.eps: 12594 bytes w21d.eps: 12278 bytes z1d.eps: 12984 bytes, is(are) stored in gzipped form as 0801.2532.tar.gz with size 46kb. The corresponding postcript file has gzipped size 180kb. Submitted from: araujoj(a)unican.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2532 or http://arXiv.org/abs/0801.2532 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2532 or in gzipped form by using subject line get 0801.2532 to: math(a)arXiv.org.

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Abstract of a paper by Yves Dutrieux Gilles Lancien
by alspach＠fourier.math.okstate.edu 19 Feb '08

19 Feb '08

This is an announcement for the paper "Isometric embeddings of compact spaces into Banach spaces" by Yves Dutrieux Gilles Lancien. Abstract: We show the existence of a compact metric space $K$ such that whenever $K$ embeds isometrically into a Banach space $Y$, then any separable Banach space is linearly isometric to a subspace of $Y$. We also address the following related question: if a Banach space $Y$ contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space $X$, does it necessarily contain a subspace isometric to $X$? We answer positively this question when $X$ is a polyhedral finite-dimensional space, $c_0$ or $\ell_1$. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46B20 Remarks: 8 pages The source file(s), dutrieux_lancien.tex: 22590 bytes, is(are) stored in gzipped form as 0801.2486.gz with size 8kb. The corresponding postcript file has gzipped size 79kb. 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/0801.2486 or http://arXiv.org/abs/0801.2486 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2486 or in gzipped form by using subject line get 0801.2486 to: math(a)arXiv.org.

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Abstract of a paper by Jarno Talponen
by alspach＠fourier.math.okstate.edu 06 Feb '08

06 Feb '08

This is an announcement for the paper "A note on the class of super reflexive almost transitive Banach spaces" by Jarno Talponen. Abstract: The class J of simultaneously almost transitive, uniformly convex and uniformly smooth Banach spaces is characterized in terms of convex-transitivity and weak geometry of the norm. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46B20 The source file(s), NoteJ.tex: 21992 bytes, is(are) stored in gzipped form as 0801.2320.gz with size 8kb. The corresponding postcript file has gzipped size 57kb. Submitted from: talponen(a)cc.helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2320 or http://arXiv.org/abs/0801.2320 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2320 or in gzipped form by using subject line get 0801.2320 to: math(a)arXiv.org.

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Abstract of a paper by Jarno Talponen
by alspach＠fourier.math.okstate.edu 06 Feb '08

06 Feb '08

This is an announcement for the paper "Operators on C_{0}(L,X) whose range does not contain c_{0}" by Jarno Talponen. Abstract: This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\to C_{0}(L,X), whose range does not contain C_{00} isomorphically, satisfies the Daugavet equality ||I+T||=1+||T||. b) Let \Gamma be a non-empty set and X, Y be Banach spaces such that X is reflexive and Y does not contain c_{0} isomorphically. Then any continuous linear operator T: c_{0}(\Gamma,X)\to Y is weakly compact. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46B28 The source file(s), dgvt_talponen.tex: 17582 bytes, is(are) stored in gzipped form as 0801.2314.gz with size 6kb. The corresponding postcript file has gzipped size 61kb. Submitted from: talponen(a)cc.helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2314 or http://arXiv.org/abs/0801.2314 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2314 or in gzipped form by using subject line get 0801.2314 to: math(a)arXiv.org.

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Abstract of a paper by Geraldo Botelho and Daniel Pellegrino
by alspach＠fourier.math.okstate.edu 06 Feb '08

06 Feb '08

This is an announcement for the paper "Absolutely summing linear operators into spaces with no finite cotype" by Geraldo Botelho and Daniel Pellegrino. Abstract: Given an infinite-dimensional Banach space $X$ and a Banach space $Y$ with no finite cotype, we determine whether or not every continuous linear operator from $X$ to $Y$ is absolutely $(q;p)$-summing for almost all choices of $p$ and $q$, including the case $p=q$. If $X$ assumes its cotype, the problem is solved for all choices of $p$ and $q$. Applications to the theory of dominated multilinear mappings are also provided. Archive classification: math.FA Mathematics Subject Classification: 47B10 Remarks: 7 pages The source file(s), Botelho-Pellegrino-BullPolish.tex: 22261 bytes, is(are) stored in gzipped form as 0801.2051.gz with size 7kb. The corresponding postcript file has gzipped size 74kb. Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2051 or http://arXiv.org/abs/0801.2051 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2051 or in gzipped form by using subject line get 0801.2051 to: math(a)arXiv.org.

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Abstract of a paper by Marius Junge
by alspach＠fourier.math.okstate.edu 06 Feb '08

06 Feb '08

This is an announcement for the paper "Noncommutative Riesz transforms I-an algebraic approach" by Marius Junge. Abstract: Riesz transforms on Rn or Riemanian manifolds are classical examples of singular integrals. In this paper we consider Riesz transforms associated to a semigroup Tt of completely positive trace preserving maps on a finite von Neumann algebra. Given a generator A of the semigroup we consider the square of the gradient Gamma(x,y)=A(x^*y)-A(x^*)y-x^*A(y) We prove un upper bound ||\Gamma(x,x)^{1/2}\|_p \le c(p) || (-\Delta)^{1/2}x ||_p under suitable assumptions. These estimates generalizes commutative results by P.A. Meyer, Bakry, Emry, Gundy, Piser. Key tools are square function inequalities obtained in joint work with C. Le Merdy and Q. Xu and new algebraic relations. As an application we obtain new examples of quantum metric spaces for discrete groups with the Haagerup property and rapid decay. Archive classification: math.OA math.FA Mathematics Subject Classification: 46L25 The source file(s), mainfile2.tex: 192365 bytes, is(are) stored in gzipped form as 0801.1873.gz with size 59kb. The corresponding postcript file has gzipped size 283kb. Submitted from: junge(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.1873 or http://arXiv.org/abs/0801.1873 or by email in unzipped form by transmitting an empty message with subject line uget 0801.1873 or in gzipped form by using subject line get 0801.1873 to: math(a)arXiv.org.

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Banach February 2008