November 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by A. G. Aksoy and J. M. Almira
by alspach＠math.okstate.edu 22 Nov '13

22 Nov '13

This is an announcement for the paper "On approximation schemes and compactness" by A. G. Aksoy and J. M. Almira. Abstract: We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introduce and show the basic properties of a new concept of compactness, which was studied by the first author in the eighties, by using a generalized concept of approximation scheme and its associated Kolmogorov numbers, which generalizes the classical concept of compactness. Archive classification: math.FA Remarks: 18 pages, submitted Submitted from: jmalmira(a)ujaen.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.2385 or http://arXiv.org/abs/1311.2385

1 0

Abstract of a paper by Rainis Haller, Johann Langemets and Mart Poldvere
by alspach＠math.okstate.edu 22 Nov '13

22 Nov '13

This is an announcement for the paper "On duality of diameter 2 properties" by Rainis Haller, Johann Langemets and Mart Poldvere. Abstract: It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and Maurey). We introduce two more versions of octahedrality, which turn out to be dual properties to the diameter 2 property and its local version (i.e., respectively, every relatively weakly open subset and every slice of the unit ball has diameter 2). We study stability properties of different types of octahedrality, which, by duality, provide easier proofs of many known results on diameter 2 properties. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Submitted from: johann.langemets(a)ut.ee The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.2177 or http://arXiv.org/abs/1311.2177

1 0

Abstract of a paper by O.I. Reinov
by alspach＠math.okstate.edu 22 Nov '13

22 Nov '13

This is an announcement for the paper "On linear operators with ${\ssize\bold s}$-nuclear adjoints: $0<{\ssize s}\le 1$" by O.I. Reinov. Abstract: If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ \, $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\to Z_0$ so that $ Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear. Archive classification: math.FA Remarks: 11 pages, AMS TeX Submitted from: orein51(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.2270 or http://arXiv.org/abs/1311.2270

1 0

Abstract of a paper by Victor Bible
by alspach＠math.okstate.edu 22 Nov '13

22 Nov '13

This is an announcement for the paper "Using boundaries to find smooth norms" by Victor Bible. Abstract: The aim of this paper is to present a tool used to find Banach spaces which have a C^{\infty} smooth equivalent norm. The hypothesis uses particular countable decompositions of certain subsets of B_{X^*}, namely boundaries. Of interest is that the main result unifies two quite well known results. In the final section, some new Corollaries are given. Archive classification: math.FA Mathematics Subject Classification: 46B03 Remarks: 11 pages Submitted from: victor.bible(a)ucdconnect.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.1408 or http://arXiv.org/abs/1311.1408

1 0

Abstract of a paper by Bernhard Hermann Haak and Markus Haase
by alspach＠math.okstate.edu 22 Nov '13

22 Nov '13

This is an announcement for the paper "Square Function Estimates and Functional Calculi" by Bernhard Hermann Haak and Markus Haase. Abstract: In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y ), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y ) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F , Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f ) for certain H-valued functions f on O. The assignment f to Phi_gamma(f ) then becomes a vectorial functional calculus, and a "square function estimate" for f simply means the boundedness of Phi_gamma(f ). In this view, all results linking square function estimates with the boundedness of a certain (usually the H-infinity) functional calculus simply assert that certain square function estimates imply other square function estimates. In the present paper several results of this type are proved in an abstract setting, based on the principles of subordination, integral representation, and a new boundedness concept for subsets of Hilbert spaces, the so-called ell-1 -frame-boundedness. These abstract results are then applied to the H-infinity calculus for sectorial and strip type operators. For example, it is proved that any strip type operator with bounded scalar H-infinity calculus on a strip over a Banach space with finite cotype has a bounded vectorial H-infinity calculus on every larger strip. Archive classification: math.FA Remarks: 49p. Submitted from: bernhard.haak(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.0453 or http://arXiv.org/abs/1311.0453

1 0