Banach

banach@mathdept.okstate.edu

2549 discussions

Kent State Informal Analysis Seminar October 27-28
by Artem Zvavitch 10 Oct '07

10 Oct '07

Dear Friends, In October 27-28, 2007 (Saturday-Sunday), the Department of Mathematical Science at Kent State University will run the famous but still very informal INFORMAL ANALYSIS SEMINAR The plan for now is to start in the morning of Saturday October 27 and finish around 3-4pm Sunday October 28 (this time it was decided to not have a break for Saturday night, it will save us and you some hotel money :) ). The list of speakers will include * Keith Ball (University College London). * Alexandre Eremenko (Purdue University). * William B. Johnson (Texas A&M University). * Fedor Nazarov (University of Wisconsin-Madison). * Andreas Seeger (University of Wisconsin-Madison). * Thomas Schlumprecht(Texas A&M University). * Vladimir Temlyakov (University of South Carolina) It would be great if you could visit Kent State and participate in the seminar! May we ask you to respond as soon as possible, so that we can gauge the need for housing, lecture room(s), etc. Please, check http://www.math.kent.edu/math/Informal-Analysis-Seminar-2007.cfm for more information. The Seminar is supported by the Department of Mathematical Sciences and NSF Focused Research Group: Fourier analytic and probabilistic methods in geometric functional analysis and convexity. Minorities, women, graduate students, and young researchers are especially encouraged to attend. Best Regards, Analysis group at Kent State!

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Abstract of a paper by Anna Maria Pelczar
by Dale Alspach 28 Sep '07

28 Sep '07

This is an announcement for the paper "Note on distortion and Bourgain $\ell_1$ index" by Anna Maria Pelczar. Abstract: The relation between different notions measuring proximity to $\ell_1$ and distortability of a Banach space is studied. The main result states that a Banach space, whose all subspaces have Bourgain $\ell_1$ index greater than $\omega^\alpha$, $\alpha<\omega_1$, contains either an arbitrary distortable subspace or an $\ell_1^\alpha$-asymptotic subspace. Archive classification: math.FA Mathematics Subject Classification: 46B20 (primary), 46B03 (secondary) Remarks: 10 pages The source file(s), distortion_bourgain.tex: 36771 bytes, is(are) stored in gzipped form as 0709.2272.gz with size 11kb. The corresponding postcript file has gzipped size 92kb. Submitted from: anna.pelczar(a)im.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.2272 or http://arXiv.org/abs/0709.2272 or by email in unzipped form by transmitting an empty message with subject line uget 0709.2272 or in gzipped form by using subject line get 0709.2272 to: math(a)arXiv.org.

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Abstract of a paper by R. M. Dudley, Sergiy Sidenko, Zuoqin Wang, and Fangyun Yang
by Dale Alspach 28 Sep '07

28 Sep '07

This is an announcement for the paper "Some classes of rational functions and related Banach spaces" by R. M. Dudley, Sergiy Sidenko, Zuoqin Wang, and Fangyun Yang. Abstract: For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded above by M and below by 1/M. Each numerator is a monic monomial of the same degree as the corresponding denominator. Then we form the Banach space of countable linear combinations of such rational functions with absolutely summable coefficients, normed by the infimum of sums of absolute values of the coefficients. We show that for rational functions whose denominators are rth powers of a specific 1+Q, or differences of two such rational functions with the same numerator, the norm is achieved by and only by the obvious combination of one or two functions respectively. We also find bounds for coefficients in partial-fraction decompositions of some specific rational functions, which in some cases are quite sharp. Archive classification: math.FA Mathematics Subject Classification: 46B99 (primary), 46B22 (secondary) Remarks: LaTex, 18 pages, no figures The source file(s), bspsrtlfncts.tex: 74856 bytes, is(are) stored in gzipped form as 0709.2449.gz with size 25kb. The corresponding postcript file has gzipped size 93kb. Submitted from: rmd(a)math.mit.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.2449 or http://arXiv.org/abs/0709.2449 or by email in unzipped form by transmitting an empty message with subject line uget 0709.2449 or in gzipped form by using subject line get 0709.2449 to: math(a)arXiv.org.

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Abstract of a paper by Tuomas Hytonen, Jan van Neerven, and Pierre Portal
by Dale Alspach 13 Sep '07

13 Sep '07

This is an announcement for the paper "Conical square functions in UMD Banach spaces" by Tuomas Hytonen, Jan van Neerven, and Pierre Portal. Abstract: We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator $A$ with certain off-diagonal bounds, such that $A$ always has a bounded $H^{\infty}$-functional calculus on these spaces. This provides a new way of proving functional calculus of $A$ on the Bochner spaces $L^p(\R^n;X)$ by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when $X=\C$, our approach gives refined $p$-dependent versions of known results. Archive classification: math.FA math.SP Mathematics Subject Classification: Primary: 46B09; Secondary: 42B25, 42B35, 46B09, 46E40, 47A60, 47F05 Remarks: 28 pages; submitted for publication The source file(s), tent/newsymbol.sty: 440 bytes tent/tent.bbl: 5616 bytes tent/tent.tex: 91867 bytes, is(are) stored in gzipped form as 0709.1350.tar.gz with size 29kb. The corresponding postcript file has gzipped size 167kb. Submitted from: J.M.A.M.vanNeerven(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.1350 or http://arXiv.org/abs/0709.1350 or by email in unzipped form by transmitting an empty message with subject line uget 0709.1350 or in gzipped form by using subject line get 0709.1350 to: math(a)arXiv.org.

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Abstract of a paper by Osman Gueler and Filiz Guertuna
by Dale Alspach 10 Sep '07

10 Sep '07

This is an announcement for the paper "The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases" by Osman Gueler and Filiz Guertuna. Abstract: A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups of convex bodies and their extremal ellipsoids. We show that if the automorphism group of a convex body K is large enough, then it is possible to determine the extremal ellipsoids CE(K) and IE(K) exactly, using either semi-infinite programming or nonlinear programming. As examples, we compute the extremal ellipsoids when the convex body K is the part of a given ellipsoid between two parallel hyperplanes, and when K is a truncated second order cone or an ellipsoidal cylinder. Archive classification: math.OC math.FA Mathematics Subject Classification: 90C34; 46B20; 90C30; 90C46; 65K10 Remarks: 36 pages The source file(s), Ellipsoid35.bbl: 8177 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0707 or http://arXiv.org/abs/0709.0707 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0707 or in gzipped form by using subject line get 0709.0707 to: math(a)arXiv.org.

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Abstract of a paper by Sorina Barza, Viktor Kolyada, and Javier Soria
by Dale Alspach 10 Sep '07

10 Sep '07

This is an announcement for the paper "Sharp constants related to the triangle inequality in Lorentz spaces" by Sorina Barza, Viktor Kolyada, and Javier Soria. Abstract: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\}, $$ where the infimum is taken over all finite representations $f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $$ ||f||_{p,s}'= \sup\left\{ \int_R fg\,d\mu: ||g||_{p',s'}=1\right\} $$ agree for all values $p,s>1$. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E30, 46B25 Remarks: 24 pages The source file(s), Norms-Constants.tex: 47398 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0647 or http://arXiv.org/abs/0709.0647 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0647 or in gzipped form by using subject line get 0709.0647 to: math(a)arXiv.org.

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Abstract of a paper by Joel A. Tropp
by Dale Alspach 05 Sep '07

05 Sep '07

This is an announcement for the paper "On the linear independence of spikes and sines" by Joel A. Tropp. Abstract: The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves methods from geometric functional analysis. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 47A11, 15A52 Remarks: 4 figures The source file(s), art/old/square-unnorm.eps: 11263 bytes, etc., is(are) stored in gzipped form as 0709.0517.tar.gz with size 344kb. The corresponding postcript file has gzipped size 173kb. The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0517 or http://arXiv.org/abs/0709.0517 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0517 or in gzipped form by using subject line get 0709.0517 to: math(a)arXiv.org.

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Abstract of a paper by Marius Junge and Quanhua Xu
by Dale Alspach 05 Sep '07

05 Sep '07

This is an announcement for the paper "Counterexamples for the convexity of certain matricial inequalities" by Marius Junge and Quanhua Xu. Abstract: In \cite{CL} Carlen and Lieb considered Minkowski type inequalities in the context of operators on a Hilbert space. More precisely, they considered the homogenous expression \[ f_{pq}(x_1,...,x_n) \lel \big(tr\big((\sum_{k=1}^n x_k^q)^{p/q}\big)\big)^{1/p} \pl \] defined for positive matrices. The concavity for $q=1$ and $p<1$ yields strong subadditivity for quantum entropy. We discuss the convexity of $f_{pq}$ and show that, contrary to the commutative case, there exists a $q_0>1$ such that $f_{1q}$ is not convex for all $1<q<q_0$. This is achieved by constructing a family of interesting channels on $2\times 2$ matrices. Archive classification: math.FA math-ph math.MP Mathematics Subject Classification: 46L25 15A48 The source file(s), cedriv.tex: 58533 bytes, is(are) stored in gzipped form as 0709.0433.gz with size 18kb. The corresponding postcript file has gzipped size 129kb. 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/0709.0433 or http://arXiv.org/abs/0709.0433 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0433 or in gzipped form by using subject line get 0709.0433 to: math(a)arXiv.org.

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Abstract of a paper by Yun Sung Choi, Han Ju Lee, and Hyun Gwi Song
by Dale Alspach 31 Aug '07

31 Aug '07

This is an announcement for the paper "Bishop's theorem and differentiability of a subspace of $C_b(K)$" by Yun Sung Choi, Han Ju Lee, and Hyun Gwi Song. Abstract: Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the G\^{a}teaux and Fr\'echet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_\delta$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikod\'ym property, we show that the set of all strong peak functions in $A_b(B_X)=\{ f\in C_b(B_X) : f|_{B_X^\circ} \mbox{ is holomorphic}\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is G\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fr\'echet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46G20; 46G25; 46B22 The source file(s), bishop-070130.tex: 87264 bytes, is(are) stored in gzipped form as 0708.4069.gz with size 25kb. The corresponding postcript file has gzipped size 157kb. Submitted from: hahnju(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.4069 or http://arXiv.org/abs/0708.4069 or by email in unzipped form by transmitting an empty message with subject line uget 0708.4069 or in gzipped form by using subject line get 0708.4069 to: math(a)arXiv.org.

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Abstract of a paper by Yun Sung Choi Kwang Hee Han Han Ju Lee
by Dale Alspach 31 Aug '07

31 Aug '07

This is an announcement for the paper "Boundaries for algebras of holomorphic functions on Banach spaces" by Yun Sung Choi Kwang Hee Han Han Ju Lee. Abstract: We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\varphi, w}$ if $\varphi$ satisfies the $\delta_2$-condition. Archive classification: math.FA Mathematics Subject Classification: 46E50; 46B20; 46B45 The source file(s), shilovboundary-final-corrected.tex: 39013 bytes, is(are) stored in gzipped form as 0708.4068.gz with size 12kb. The corresponding postcript file has gzipped size 102kb. Submitted from: hahnju(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.4068 or http://arXiv.org/abs/0708.4068 or by email in unzipped form by transmitting an empty message with subject line uget 0708.4068 or in gzipped form by using subject line get 0708.4068 to: math(a)arXiv.org.

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