Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Han Ju Lee
by Dale Alspach 19 May '07

19 May '07

This is an announcement for the paper "Strong peak points and denseness of strong peak functions" by Han Ju Lee. Abstract: Let $C_b(K)$ be the set of all bounded continuous (real or complex) functions on a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using the variational method, it is shown that the set of all strong peak functions in $A$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we show that if $X$ is a locally uniformly convex, complex Banach space, then the set of all strong peak functions in $\mathcal{A}(B_X)$ is a dense $G_\delta$ subset. Moreover if $X$ is separable, smooth and locally uniformly convex, then the set of all norm and numerical strong peak functions in $\mathcal{A}_u(B_X:X)$ is a dense $G_\delta$ subset. In case that a set of uniformly strongly exposed points of a (real or complex) Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$ for some $n\ge 1$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense, in particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46G20, 46G25, 46B22 The source file(s), variationalmethod-2007-04-15.tex: 25864 bytes, is(are) stored in gzipped form as 0705.2650.gz with size 8kb. The corresponding postcript file has gzipped size 75kb. 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/0705.2650 or http://arXiv.org/abs/0705.2650 or by email in unzipped form by transmitting an empty message with subject line uget 0705.2650 or in gzipped form by using subject line get 0705.2650 to: math(a)arXiv.org.

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Abstract of a paper by Jarno Talponen
by Dale Alspach 19 May '07

19 May '07

This is an announcement for the paper "Convex-transitive characterizations of Hilbert spaces" by Jarno Talponen. Abstract: In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46C15 The source file(s), amsct2.tex: 89202 bytes, is(are) stored in gzipped form as 0705.2526.gz with size 24kb. The corresponding postcript file has gzipped size 142kb. 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/0705.2526 or http://arXiv.org/abs/0705.2526 or by email in unzipped form by transmitting an empty message with subject line uget 0705.2526 or in gzipped form by using subject line get 0705.2526 to: math(a)arXiv.org.

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History of Banach Spaces and Linear Operators
by Dale Alspach 03 May '07

03 May '07

This is an announcement of the publication of the book History of Banach Spaces and Linear Operators by Albrecht Pietsch The table of contents and preface can be viewed here: http://www.math.okstate.edu/~alspach/banach/pietsch-history.pdf

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Abstract of a paper by Daniel Freeman
by Dale Alspach 03 May '07

03 May '07

This is an announcement for the paper "Weakly null sequences with upper estimates" by Daniel Freeman. Abstract: We prove that if $(v_i)$ is a normalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_i)$. This extends a result of Knaust and Odell, who proved this for the cases in which $(v_i)$ is the standard basis for $\ell_p$ or $c_0$. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46B03, 46B10 Remarks: 21 pages The source file(s), FreemanUpEst.tex, is(are) stored in gzipped form as 0705.0218.gz with size 20kb. The corresponding postcript file has gzipped size 146kb. Submitted from: freeman(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0705.0218 or http://arXiv.org/abs/0705.0218 or by email in unzipped form by transmitting an empty message with subject line uget 0705.0218 or in gzipped form by using subject line get 0705.0218 to: math(a)arXiv.org.

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Abstract of a paper by Florent Baudier
by Dale Alspach 17 Apr '07

17 Apr '07

This is an announcement for the paper "Metrical characterization of super-reflexivity and linear type of Banach spaces" by Florent Baudier. Abstract: We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric. Archive classification: Mathematics Subject Classification: 46B20; 51F99 Remarks: to appear in Archiv der Mathematik The source file(s), , is(are) stored in gzipped form as 0704.1955.gz with size 8kb. The corresponding postcript file has gzipped size 78kb. Submitted from: florent.baudier(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/ or http://arXiv.org/abs/ or by email in unzipped form by transmitting an empty message with subject line uget or in gzipped form by using subject line get to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 17 Apr '07

17 Apr '07

This is an announcement for the paper "Even infinite dimensional real Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: This article is a continuation of a paper of the first author \cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from \cite{F} and $C(K)$ examples due to Plebanek \cite{P}. We extend results of \cite{F} relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of \cite{F} about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis \cite{AM} provide examples of essentially incomparable complex structures which are not totally incomparable. Archive classification: Mathematics Subject Classification: 46B03; 47A53. Remarks: 22 pages The source file(s), , is(are) stored in gzipped form as 0704.1459.gz with size 16kb. The corresponding postcript file has gzipped size 85kb. Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/ or http://arXiv.org/abs/ or by email in unzipped form by transmitting an empty message with subject line uget or in gzipped form by using subject line get to: math(a)arXiv.org.

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Abstract of a paper by Jesus Araujo
by Dale Alspach 10 Apr '07

10 Apr '07

This is an announcement for the paper "Examples and counterexamples of type I isometric shifts" by Jesus Araujo. Abstract: We provide examples of nonseparable spaces $X$ for which $C(X)$ admits an isometric shift of type I, which solves in the negative a problem proposed by Gutek {\em et al.} (J. Funct. Anal. {\bf 101} (1991), 97-119). We also give two independent methods for obtaining separable examples. The first one allows us in particular to construct examples with infinitely many nonhomeomorphic components in a subset of the Hilbert space $\ell^2$. The second one applies for instance to sequences adjoined to any $n$-dimensional compact manifold (for $n \ge 2$) or to the Sierpi\'nski curve. The combination of both techniques lead to different examples involving a convergent sequence adjoined to the Cantor set: one method for the case when the sequence converges to a point in the Cantor set, and the other one for the case when it converges outside. Archive classification: Functional Analysis; General Topology Mathematics Subject Classification: Primary 47B38; Secondary 46E15, 47B33, 47B37, 54D65, 54H20 Remarks: 41 pages. No figures. AMS-LaTeX The source file(s), shiftnum86.tex: 124237 bytes, is(are) stored in gzipped form as 0703892.gz with size 34kb. The corresponding postcript file has gzipped size 210kb. Submitted from: araujoj(a)unican.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0703892 or http://arXiv.org/abs/math.FA/0703892 or by email in unzipped form by transmitting an empty message with subject line uget 0703892 or in gzipped form by using subject line get 0703892 to: math(a)arXiv.org.

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Abstract of a paper by Boris Rubin
by Dale Alspach 10 Apr '07

10 Apr '07

This is an announcement for the paper "Intersection bodies and generalized cosine transforms" by Boris Rubin. Abstract: Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. We review some known facts and give them new proofs. The main focus is interrelation between generalized cosine transforms of different kinds and their application to investigation of certain family of intersection bodies, which we call lambda-intersection bodies. The latter include k-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restriction of the spherical Radon transforms and the generalized cosine transforms onto lower dimensional subspaces preserves their integral-geometric structure. We apply this result to the study of sections of lambda-intersection bodies. A number of new characterizations of this class of bodies and examples are given. Archive classification: Mathematics Subject Classification: 44A12; 52A38 Remarks: 36 pages The source file(s), , is(are) stored in gzipped form as 0704.0061.gz with size 31kb. The corresponding postcript file has gzipped size 195kb. Submitted from: borisr(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/ or http://arXiv.org/abs/ or by email in unzipped form by transmitting an empty message with subject line uget or in gzipped form by using subject line get to: math(a)arXiv.org.

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Abstract of a paper by Dumitru Popa
by Dale Alspach 22 Mar '07

22 Mar '07

This is an announcement for the paper "Khinchin's inequality, Dunford--Pettis and compact operators on the space $\pmb{C([0,1],X)}$" by Dumitru Popa. Abstract: We prove that if $X,Y$ are Banach spaces, $\Omega$ a compact Hausdorff space and $U\hbox{\rm :}\ C(\Omega,X)\rightarrow Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(\Sigma) \subseteq DP(X,Y)$ is an uniformly Dunford--Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_{0}$ or $l_{p}$, ($1\leq p<\infty$) be Dunford--Pettis and/or compact operators, in which, Khinchin's inequality plays an important role. Archive classification: Functional Analysis Mathematics Subject Classification: 46B28; 47A80; 47B10 Remarks: 18 pages The source file(s), mat01.cls: 37299 bytes, mathtimy.sty: 20 bytes, pm2710new.tex: 66481 bytes, is(are) stored in gzipped form as 0703626.tar.gz with size 24kb. The corresponding postcript file has gzipped size 76kb. Submitted from: dpopa(a)univ-ovidius.ro The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0703626 or http://arXiv.org/abs/math.FA/0703626 or by email in unzipped form by transmitting an empty message with subject line uget 0703626 or in gzipped form by using subject line get 0703626 to: math(a)arXiv.org.

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Abstract of a paper by Andrea Colesanti
by Dale Alspach 21 Mar '07

21 Mar '07

This is an announcement for the paper "From the Brunn-Minkowski inequality to a class of Poincar\'e type inequalities" by Andrea Colesanti. Abstract: We present an argument which leads from the Brunn-Minkowski inequality to a Poincare' type inequality on the boundary of convex bodies with smooth boundary and positive Gauss curvature. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A20; 26D10 Remarks: 9 pages The source file(s), testo.tex: 21763 bytes, is(are) stored in gzipped form as 0703584.gz with size 7kb. The corresponding postcript file has gzipped size 93kb. Submitted from: colesant(a)math.unifi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0703584 or http://arXiv.org/abs/math.FA/0703584 or by email in unzipped form by transmitting an empty message with subject line uget 0703584 or in gzipped form by using subject line get 0703584 to: math(a)arXiv.org.

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