Banach

banach@mathdept.okstate.edu

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by Daniel LI 01 Apr '05

01 Apr '05

Le volume 12 de la collection "Cours Specialises" est a votre disposition. INTRODUCTION A L'ETUDE DES ESPACES DE BANACH ANALYSE ET PROBABILITES Daniel Li, Herve Queffelec xxiv+627 pages Ce livre est consacre a l'etude des espaces de Banach, en mettant l'accent sur les liens avec l'Analyse classique, l'Analyse Harmonique, et les Probabilites. Seules des connaissances usuelles d'Analyse Fonctionnelle de niveau Maitrise sont requises, l'etude etant prise a son debut. Elle est progressivement developpee de facon approfondie, presentant plusieurs resultats fondamentaux obtenus dans la periode 1950--2000: Theoreme de Grothendieck, Theoreme de Dvoretzky, Theoreme de dichotomie de Rosenthal, Theoreme de dichotomie de Gowers, etc., avec certaines de leurs applications. This book is devoted to the study of Banach spaces, with emphasis on the connections with classical Analysis, Harmonic Analysis and Probability Theory. It can be tackled by beginning graduates: the study is taken at its beginning, and then worked out thoroughly, presenting several fundamental results which were obtained during the period 1950--2000: Grothendieck's Theorem, Dvoretzky's Theorem, Rosenthal's dichotomy Theorem, Gowers's dichotomy Theorem, etc., with some of their applications. Prix public (pour chaque volume) : 72 euro + frais de port (France : 5 euro, Europe : 6 euro, Hors Europe : 7 euro) Prix membre (pour chaque volume) : 51 euro + frais de port (France : 5 euro, Europe : 6 euro, Hors Europe : 7 euro) Vous pouvez vous procurer ces volumes en le commandant a la cellule de diffusion de Marseille : Maison de la SMF, BP 67, 13274 Marseille cedex 09, Tel : (33) 04 91 26 74 64, Fax : (33) 04 91 41 17 57, email : smf(a)smf.univ-mrs.fr, url : http://smf.emath.fr/, ou en passant le chercher au secretariat de la SMF a Paris (IHP, 11 rue Pierre et Marie Curie 75005 Paris). Vous pouvez aussi passer directement votre commande par l'intermediaire du serveur a l'adresse suivante : http://smf.emath.fr/ Daniel Li Université d'Artois Laboratoire de Mathématiques de Lens (LML) Faculté des Sciences Jean Perrin rue Jean Souvraz, SP 18 62307 LENS Cedex Tel +33 (0)3 21 79 17 22 Fax +33 (0)3 21 79 17 29 daniel.li(a)euler.univ-artois.fr

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Abstract of a paper by Madjid Mirzavaziri and Mohammad Sal Moslehian
by Dale Alspach 29 Mar '05

29 Mar '05

This is an announcement for the paper "Parallelogram norm" by Madjid Mirzavaziri and Mohammad Sal Moslehian. Abstract: Replacing the triangle inequality by \|x+y\|^2\leq 2(\|x\|^2 + \|y\|^2) in the definition of norm we obtain the notion of parallelogram norm. We establish that every parallelogram norm is a norm in the usual sense. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 46C05 Remarks: 3 pages The source file(s), Paral1.tex: 4582 bytes, is(are) stored in gzipped form as 0503616.gz with size 2kb. The corresponding postcript file has gzipped size 27kb. Submitted from: msalm(a)math.um.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503616 or http://arXiv.org/abs/math.FA/0503616 or by email in unzipped form by transmitting an empty message with subject line uget 0503616 or in gzipped form by using subject line get 0503616 to: math(a)arXiv.org.

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Abstract of a paper by Mark Rudelson and Roman Vershynin
by Dale Alspach 23 Mar '05

23 Mar '05

This is an announcement for the paper "Sampling from large matrices: an approach through geometric functional analysis" by Mark Rudelson and Roman Vershynin. Abstract: We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix. This improves known algorithms for computing low-rank approximations of large matrices. We also estimate norms of random submatrices of A. This yields an improved approximation algorithm for all MAX-2CSP problems (which includes MAX-CUT and other graph problems). Our results are essentially dimension-free; the picture is only controlled by the norms of the matrix and not by its size or rank. We use methods of Probability in Banach spaces, in particular the law of large numbers for random operators. Archive classification: Functional Analysis; Numerical Analysis Mathematics Subject Classification: 15A60, 68W20, 15A18 The source file(s), rv-random-submatrices.tex: 50699 bytes, is(are) stored in gzipped form as 0503442.gz with size 16kb. The corresponding postcript file has gzipped size 82kb. Submitted from: vershynin(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503442 or http://arXiv.org/abs/math.FA/0503442 or by email in unzipped form by transmitting an empty message with subject line uget 0503442 or in gzipped form by using subject line get 0503442 to: math(a)arXiv.org.

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Abstract of a paper by V.Yaskin and M.Yaskina
by Dale Alspach 22 Mar '05

22 Mar '05

This is an announcement for the paper "Centroids and comparison of volumes" by V.Yaskin and M.Yaskina. Abstract: For $-1<p<1$ we introduce the concept of a polar $p$-centroid body ${\Gamma^*_p K}$ of a star body $K$. We consider the question of whether ${\Gamma^*_p K}\subset {\Gamma^*_p L}$ implies $\mathrm{vol}(L)\le \mathrm{vol}(K).$ Our results extend the studies by Lutwak in the case $p=1$ and Grinberg, Zhang in the case $p> 1$. Archive classification: Functional Analysis Mathematics Subject Classification: 52Axx Remarks: 18 pages The source file(s), centr.tex: 51970 bytes, is(are) stored in gzipped form as 0503290.gz with size 13kb. The corresponding postcript file has gzipped size 71kb. Submitted from: yaskinv(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503290 or http://arXiv.org/abs/math.FA/0503290 or by email in unzipped form by transmitting an empty message with subject line uget 0503290 or in gzipped form by using subject line get 0503290 to: math(a)arXiv.org.

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Abstract of a paper by V.Yaskin
by Dale Alspach 22 Mar '05

22 Mar '05

This is an announcement for the paper "A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space" by V.Yaskin. Abstract: The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if $k>3$. The problem is still open for $k=2,3$. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space $\mathbb{H}^n$. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A55, 52A20, 46B20 Remarks: 12 pages, 2 figures The source file(s), LDHBP.tex: 70816 bytes, pic04.eps: 9457 bytes, pic06.eps: 9542 bytes, is(are) stored in gzipped form as 0503289.tar.gz with size 25kb. The corresponding postcript file has gzipped size 59kb. Submitted from: yaskinv(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503289 or http://arXiv.org/abs/math.FA/0503289 or by email in unzipped form by transmitting an empty message with subject line uget 0503289 or in gzipped form by using subject line get 0503289 to: math(a)arXiv.org.

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Abstract of a paper by Miguel Martin, Javier Meri and Rafael Paya
by Dale Alspach 09 Mar '05

09 Mar '05

This is an announcement for the paper "On the intrinsic and the spatial numerical range" by Miguel Martin, Javier Meri and Rafael Paya. Abstract: For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a bounded linear operator $T:X\longrightarrow Y$ such that $\ecc W(T)\neq V(T)$. We also show that, up to renormig, for every non-reflexive Banach space $Y$, one can find a closed subspace $X$ and a bounded linear operator $T\in L(X,Y)$ such that $\ecc W(T)\neq V(T)$. Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobas property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 47A12 Remarks: 12 pages The source file(s), MartinMeriPaya.tex: 40725 bytes, is(are) stored in gzipped form as 0503076.gz with size 13kb. The corresponding postcript file has gzipped size 70kb. Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503076 or http://arXiv.org/abs/math.FA/0503076 or by email in unzipped form by transmitting an empty message with subject line uget 0503076 or in gzipped form by using subject line get 0503076 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 17 Feb '05

17 Feb '05

This is an announcement for the paper "Weak OH-type 2 and weak OH-cotype 2 of operator spaces" by Hun Hee Lee. Abstract: Recently, OH-type and OH-cotype of operator spaces, an operator space version of type and cotype, were introduced and investigated by the author. In this paper we define weak OH-type 2 (resp. weak OH-cotype 2) of operator spaces, which lies strictly between OH-type 2 (resp. OH-cotype 2) and OH-type $p$ for all $1 \leq p < 2$. (resp. OH-cotype $q$ for all $2< q <= \infty$) This is an analogue of weak type 2 and weak cotype 2 in Banach space case, so we develop analogous theory focusing on the local properties of spaces with such conditions. Archive classification: Functional Analysis; Operator Algebras Remarks: 21 pages The source file(s), WeakOH.tex: 55124 bytes, is(are) stored in gzipped form as 0502337.gz with size 15kb. The corresponding postcript file has gzipped size 85kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502337 or http://arXiv.org/abs/math.FA/0502337 or by email in unzipped form by transmitting an empty message with subject line uget 0502337 or in gzipped form by using subject line get 0502337 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "Eigenvalues of completely nuclear maps and completely bounded projection constants" by Hun Hee Lee. Abstract: We investigate the distribution of eigenvalues of completely nuclear maps on an operator space. We prove that eigenvalues of completely nuclear maps are square-summable in general and summable if the underlying operator space is Hilbertian and homogeneous. Conversely, if eigenvalues are summable for all completely nuclear maps, then every finite dimensional subspace of the underlying operator space is uniformly completely complemented. As an application we consider an estimate of completely bounded projection constants of $n$-dimensional operator spaces. Archive classification: Functional Analysis; Operator Algebras Remarks: 10 pages The source file(s), EigenComNuclear.tex: 27465 bytes, is(are) stored in gzipped form as 0502335.gz with size 9kb. The corresponding postcript file has gzipped size 55kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502335 or http://arXiv.org/abs/math.FA/0502335 or by email in unzipped form by transmitting an empty message with subject line uget 0502335 or in gzipped form by using subject line get 0502335 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "OH-type and OH-cotype of operator spaces and completely summing maps" by Hun Hee Lee. Abstract: The definition and basic properties of OH-type and OH-cotype of operator spaces are given. We prove that every bounded linear map from C(K) into OH-cotype q (2<= q < infinity) space (including most of commutative L_q-spaces) for a compact set K satisfies completely (q,2)-summing property, a noncommutative analogue of absolutely (q,2)-summing property. At the end of this paper, we observe that ``OH-cotype 2" is equivalent to the previous definition of ``OH-cotype 2" of G. Pisier. Archive classification: Functional Analysis; Operator Algebras Remarks: 17 pages The source file(s), OH-typecotype.tex: 46265 bytes, is(are) stored in gzipped form as 0502302.gz with size 13kb. The corresponding postcript file has gzipped size 80kb. Submitted from: hunmada(a)hanmail.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502302 or http://arXiv.org/abs/math.FA/0502302 or by email in unzipped form by transmitting an empty message with subject line uget 0502302 or in gzipped form by using subject line get 0502302 to: math(a)arXiv.org.

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Abstract of a paper by Mark Rudelson and Roman Vershynin
by Dale Alspach 16 Feb '05

16 Feb '05

This is an announcement for the paper "Geometric approach to error correcting codes and reconstruction of signals" by Mark Rudelson and Roman Vershynin. Abstract: We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes. Archive classification: Functional Analysis; Combinatorics Mathematics Subject Classification: 46B07; 94B75, 68P30, 52B05 Remarks: 17 pages, 3 figures The source file(s), ecc.tex: 50560 bytes, ecc1.eps: 4526 bytes, ecc2.eps: 17097 bytes, ecc3.eps: 4645 bytes, is(are) stored in gzipped form as 0502299.tar.gz with size 23kb. The corresponding postcript file has gzipped size 84kb. Submitted from: vershynin(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0502299 or http://arXiv.org/abs/math.FA/0502299 or by email in unzipped form by transmitting an empty message with subject line uget 0502299 or in gzipped form by using subject line get 0502299 to: math(a)arXiv.org.

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