Banach February 2012

banach@mathdept.okstate.edu

3 participants
16 discussions

Abstract of a paper by Emanuele Casini, Enrico Miglierina, Ioannis A. Polyrakis, and Foivos Xanthos
by alspach＠math.okstate.edu 16 Feb '12

16 Feb '12

This is an announcement for the paper "Reflexive Cones" by Emanuele Casini, Enrico Miglierina, Ioannis A. Polyrakis, and Foivos Xanthos. Abstract: Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given. Archive classification: math.FA Mathematics Subject Classification: 46B10, 46B20, 46B40, 46B42 Remarks: 23 pages Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1201.4927 or http://arXiv.org/abs/1201.4927

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Abstract of a paper by V. Mykhaylyuk, M. Popov and B. Randrianantoanina
by alspach＠math.okstate.edu 16 Feb '12

16 Feb '12

This is an announcement for the paper "On Enflo and narrow operators acting on $L_p$" by V. Mykhaylyuk, M. Popov and B. Randrianantoanina. Abstract: The paper is devoted to proofs of the following three results. Theorem A. For $1 < p < 2$ every non-Enflo operator $T$ on $L_p$ is narrow. Theorem B. For $1 < p < 2$ every operator $T$ on $L_p$ which is unbounded from below on $L_p(A)$, $A \subseteq [0,1]$, by means of function having a ``gentle'' growth, is narrow. Theorem C. For $2 < p, r < \infty$ every operator $T: L_p\rightarrow\ell_r$ is narrow. Theorem A was mentioned by Bourgain in 1981, as a result that can be deduced from the proof of a related result in Johnson-Maurey-Schechtman-Tzafriri's book, but the proof from there needed several modifications. Theorems B and C are new results. We also discuss related open problems. Archive classification: math.FA Mathematics Subject Classification: Primary 47B07, secondary 47B38, 46B03 Submitted from: randrib(a)muohio.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1201.4041 or http://arXiv.org/abs/1201.4041

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Abstract of a paper by Daniel Pellegrino and Juan B. Seoane-Sepulveda
by alspach＠math.okstate.edu 16 Feb '12

16 Feb '12

This is an announcement for the paper "On the hypercontractivity of the polynomial Bohnenblust--Hille inequality" by Daniel Pellegrino and Juan B. Seoane-Sepulveda. Abstract: Recently, it was proved that the polynomial Bohnenblust--Hille inequality is hypercontractive, i.e., there is a constant $C>1$ (from now on called constant of hypercontractivity) so that $\frac{D_{m}}{D_{m-1}}=C$ for every $m$, where $D_{m}$ are constants satisfying the polynomial Bohnenblust--Hille inequality. For the case of multilinear mappings a recent result shows that $\lim _{m\rightarrow\infty}\frac{C_{m}}{C_{m-1}}=1$, where $C_{m}$ are constants satisfying the multilinear Bohnenblust--Hille inequality. So it is natural to wonder if there exist constants $D_{m}$'s such that $\lim_{m\rightarrow\infty}\frac{D_{m}% }{D_{m-1}}=1$. In this note we provide lower estimates for the polynomial Bohnenblust--Hille inequality with strong numerical evidence supporting that it is not possible to obtain such $D_{m}.$ Besides the qualitative information, and to the best of our knowledge, this is the first time in which non-trivial lower bounds for the constants of the polynomial Bohnenblust--Hille inequality are presented. We also show that the constant of hypercontractivity $C$ is so that $1.1542\leq C\leq1.8529$, providing as well explicit formulae to improve the lower estimate $1.1542.$ It is our belief that variations of the ideas introduced in this paper can be used for the search of the optimal constants for the polynomial Bohnenblust--Hille inequality. Archive classification: math.FA Remarks: 2 figures Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1201.3873 or http://arXiv.org/abs/1201.3873

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Abstract of a paper by Gareth Speight
by alspach＠math.okstate.edu 16 Feb '12

16 Feb '12

This is an announcement for the paper "Surfaces meeting porous sets in positive measure" by Gareth Speight. Abstract: Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable this leads to a decomposition of X into a countable union of directionally porous sets and a set which is null on residually many C^1 surfaces of dimension n. This is of interest in the study of certain classes of null sets used to investigate differentiability of Lipschitz functions on Banach spaces. Archive classification: math.FA math.CA math.MG Mathematics Subject Classification: 28A75, 46T99, 46G99 Submitted from: G.Speight(a)Warwick.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1201.2376 or http://arXiv.org/abs/1201.2376

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 16 Feb '12

16 Feb '12

This is an announcement for the paper "Absolute sums of Banach spaces and some geometric properties related to rotundity and smoothness" by Jan-David Hardtke. Abstract: We study the notions of acs, luacs and uacs Banach spaces which were introduced by V. Kadets et al. in 2000 and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under taking absolute sums as well. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 42 pages, 8 figures Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1201.2300 or http://arXiv.org/abs/1201.2300

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Postdoctoral Fellowship in BESANCON, France
by Stanislaw J. Szarek 02 Feb '12

02 Feb '12

Title: Postdoctoral Research Fellowship in Functional Analysis in Besançon, France Period: September 1, 2012 to August 31, 2013 Deadline for application: May 1, 2012. We are now accepting applications for a postdoctoral research fellowship (without teaching duty) for the academic year 2012-2013 (starting date: Sept. 1,2012) in the framework of the ANR project OSQPI (Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information). We are looking for applicants who received their Ph.D. recently (or will receive it until August 2012). The fellow is expected to carry out a research project on the topics of the ANR project OSQPI (operator spaces, noncommutative Lp spaces, noncommutative harmonic analysis, quantum probability, and their applications in quantum information) at the Laboratoire de Mathématiques de Besançon (Université de Franche-Comté). Part of the program could also be carried out at partner institutions in Paris, Lyon, or Toulouse. The fellowship provides a salary of about 1.800 euro per month after taxes. For more details please contact quanhua.xu(a)univ-fcomte.fr Applications should be sent to quanhua.xu(a)univ-fcomte.fr

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Banach February 2012