August 2015 - Banach - mathdept.okstate.edu

Abstract of a paper by Bruno de Mendonca Braga
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Duality on Banach spaces and a Borel parametrized version of Zippin's theorem" by Bruno de Mendonca Braga. Abstract: Let SB be the standard coding for separable Banach spaces as subspaces of $C(\Delta)$. In these notes, we show that if $\mathbb{B} \subset \text{SB}$ is a Borel subset of spaces with separable dual, then the assignment $X \mapsto X^*$ can be realized by a Borel function $\mathbb{B}\to \text{SB}$. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem $1$). Also, we prove a Borel parametrized version of Zippin's theorem, i.e., we prove that there exists $Z \in \text{SB}$ and a Borel function that assigns for each $X \in \mathbb{B}$ an isomorphic copy of $X$ inside of $Z$ (Theorem $5$). Archive classification: math.FA Submitted from: demendoncabraga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.02066 or http://arXiv.org/abs/1508.02066

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Abstract of a paper by Jan Rozendaal
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Functional calculus for $C_{0}$-groups using (co)type" by Jan Rozendaal. Abstract: We study the functional calculus properties of generators of $C_{0}$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-\mathrm{i}A$ generate a $C_{0}$-group on a Banach space $X$ with type $p\in[1,2]$ and cotype $q\in[2,\infty)$. Then $A$ has a bounded $\mathcal{H}^{\infty}$-calculus from $\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e.\ $f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. %Hence $A$ has a bounded calculus for the class of bounded holomorphic functions which decay polynomially of order $\alpha>\frac{1}{p}-\frac{1}{q}$ at infinity. Under additional geometric assumptions, satisfied by $\mathrm{L}^{p}$-spaces, we cover the case $\alpha=\frac{1}{p}-\frac{1}{q}$. As a corollary of our main theorem, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded $\mathcal{H}^{\infty}$-calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for $C_{0}$-groups. We extend our results to $R$-bounded operator-valued calculi, and we give an application to the theory of rational approximation of $C_{0}$-groups. Archive classification: math.FA math.NA Mathematics Subject Classification: Primary 47A60, Secondary 47D03, 46B20, 42A45 Remarks: 25 pages Submitted from: janrozendaalmath(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.02036 or http://arXiv.org/abs/1508.02036

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Abstract of a paper by Bruno de Mendonca Braga
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "On the complexity of some inevitable classes of separable Banach" by Bruno de Mendonca Braga. Abstract: In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [Go], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [FR], every Banach space either contains a minimal subspace or a continuously tight subspace. In these notes, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes. Archive classification: math.FA Submitted from: demendoncabraga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.01961 or http://arXiv.org/abs/1508.01961

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Abstract of a paper by Bruno de Mendonca Braga
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "On the complexity of some classes of Banach spaces and" by Bruno de Mendonca Braga. Abstract: These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks opera- tors is complemented in its bounded operators, are all non Borel in SB. At last, we give several applications of those results to non-universality results. Archive classification: math.FA Submitted from: demendoncabraga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.01960 or http://arXiv.org/abs/1508.01960

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Abstract of a paper by Ryan M Causey
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "An ordinal index characterizing weak compactness of operators" by Ryan M Causey. Abstract: We introduce an ordinal index which characterizes weak compactness of operators between Banach spaces. We study when classes consisting of operators having bounded index form a closed ideal, the distinctness of the classes, and the descriptive set theoretic properties of this index. Archive classification: math.FA Submitted from: CAUSEYRM(a)mailbox.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.02065 or http://arXiv.org/abs/1508.02065

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Abstract of a paper by T. Figiel and W. B. Johnson
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "The Dual Form of the Approximation Property for a Banach Space and a Subspace" by T. Figiel and W. B. Johnson. Abstract: Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. The main result is an easy to apply dual formulation of this property. Applications are given to three space properties; in particular, if X has the approximation property and its subspace Y is script L-infinity, then X/Y has the approximation property. Archive classification: math.FA Mathematics Subject Classification: Primary: 46B20, 46B28 Submitted from: johnson(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.01212 or http://arXiv.org/abs/1508.01212

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Abstract of a paper by Sergey Astashkin
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Rademacher functions in weighted symmetric spaces" by Sergey Astashkin. Abstract: The closed span of Rademacher functions is investigated in the weighted spaces X(w), where X is a symmetric space on [0,1] and w is a positive measurable function on [0,1]. By using the notion and properties of the Rademacher multiplicator space of a symmetric space, we give a description of the weights w for which the Rademacher orthogonal projection is bounded in X(w). Archive classification: math.FA Mathematics Subject Classification: 46E30 (Primary), 46B20, 46B42 (Secondary) Remarks: 15 pages Submitted from: astash(a)samsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.00734 or http://arXiv.org/abs/1508.00734

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Abstract of a paper by Bernardo Cascales, Jose Orihuela and Antonio Perez
by alspach＠pelczynski.math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "One side James' Compactness Theorem" by Bernardo Cascales, Jose Orihuela and Antonio Perez. Abstract: We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, \omega^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum on $B$, then $A$ and $B$ are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 46A50, 46B50 Remarks: 18 pages Submitted from: antonio.perez7(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.00496 or http://arXiv.org/abs/1508.00496

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Abstract of a paper by David Edmunds, Amiran Gogatishvili, Tengiz Kopaliani and Nino Samashvili
by alspach＠pelczynski.math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Some $s$-numbers of an integral operator of Hardy type in Banach function spaces" by David Edmunds, Amiran Gogatishvili, Tengiz Kopaliani and Nino Samashvili. Abstract: Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\,\,\,x\in(a,b)\,\,(-\infty<a<b<+\infty) $$ and mapping a Banach function space $E$ to itself. We investigate some geometrical properties of $E$ for which $$ C_{1}\int_{a}^{b}u(x)v(x)dx \leq\limsup\limits_{n\rightarrow\infty}ns_{n}(T) \leq \limsup\limits_{n\rightarrow\infty}ns_{n}(T)\leq C_{2}\int_{a}^{b}u(x)v(x)dx $$ under appropriate conditions on $u$ and $v.$ The constants $C_{1},C_{2}>0$ depend only on the space $E.$ Archive classification: math.FA math.AP math.CA Mathematics Subject Classification: 35P30, 46E30, 46E35, 47A75 47B06, 47B10, 47B40, 47G10 Submitted from: gogatish(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.08854 or http://arXiv.org/abs/1507.08854

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Abstract of a paper by Richard Skillicorn
by alspach＠pelczynski.math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "The uniqueness-of-norm problem for Calkin algebras" by Richard Skillicorn. Abstract: We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989). Archive classification: math.FA Submitted from: r.skillicorn(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.08118 or http://arXiv.org/abs/1507.08118

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