Banach September 2012

banach@mathdept.okstate.edu

1 participants
43 discussions

Abstract of a paper by H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, and Niels Jakob Laustsen
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Maximal left ideals of the Banach algebra of bounded operators on a Banach space" by H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, and Niels Jakob Laustsen. Abstract: We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach space $E$: (I) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form $\{ T\in\mathscr{B}(E) : Tx = 0\}$ (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (II) is positive for many, but not all, Banach spaces. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 47L10, 46H10, Secondary 47L20 Submitted from: t.kania(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.4762 or http://arXiv.org/abs/1208.4762

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Abstract of a paper by Tomasz Kochanek
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Stability of vector measures and twisted sums of Banach spaces" by Tomasz Kochanek. Abstract: A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a~constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $\nu\colon\mathcal F\to X$ satisfying $$\|\nu(A\cup B)-\nu(A)-\nu(B)\|\leq 1\quad\mbox{for disjoint }A,B\in\mathcal F,$$there is a~vector measure $\mu\colon\mathcal F\to X$ with $\|\nu(A)-\mu(A)\|\leq v$ for all $A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal F$ of cardinality less than some fixed cardinal number $\kappa$, then we say that $X$ has the $\kappa$-$\mathsf{SVM}$ property. The least cardinal $\kappa$ for which $X$ does not have the $\kappa$-$\mathsf{SVM}$ property (if it exists) is called the $\mathsf{SVM}$ character of $X$. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine $\mathsf{SVM}$ characters for many classical Banach spaces. We also discuss connections between the $\kappa$-$\mathsf{SVM}$ property, $\kappa$-injectivity and the `three-space' problem. Archive classification: math.FA Mathematics Subject Classification: Primary 28B05, 46G10, 46B25, Secondary 46B03 Submitted from: t.kania(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.4755 or http://arXiv.org/abs/1208.4755

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Abstract of a paper by Svante Janson and Sten Kaijser
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Higher moments of Banach space valued random variables" by Svante Janson and Sten Kaijser. Abstract: We define the $k$:th moment of a Banach space valued random variable as the expectation of its $k$:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several sections are devoted to results in special Banach spaces, including Hilbert spaces, $C(K)$ and $D[0,1]$. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in $D[0,1]$ that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix. Archive classification: math.PR math.FA Mathematics Subject Classification: 60B11, 46G10 Remarks: 110 pages Submitted from: svante.janson(a)math.uu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.4272 or http://arXiv.org/abs/1208.4272

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Abstract of a paper by Peter G. Casazza
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "The simplified version of the Spielman and Sristave algorithm for proving the Bourgain-Tzafriri restricted invertiblity theorem" by Peter G. Casazza. Abstract: By giving up the best constants, we will see that the original argument of Spielman and Sristave for proving the Bourgain-Tzafriri Restricted Invertibility Theorem \cite{SS} still works - and is much simplier than the final version. We do not intend on publishing this since it is their argument with just a trivial modification, but we want to make it available to the mathematics community since several people have requested it already. Archive classification: math.FA Submitted from: casazzap(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.4013 or http://arXiv.org/abs/1208.4013

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Abstract of a paper by Erik Talvila
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "The $L^p$ primitive integral" by Erik Talvila. Abstract: For each $1\leq p<\infty$ a space of integrable Schwartz distributions, $L{\!}'^{\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L{\!}'^{\,p}$ such that $f$ is the distributional derivative of $F\in L^p$ then the integral is defined as $\int^\infty_{-\infty} fG=-\int^\infty_{-\infty} F(x)g(x)\,dx$, where $g\in L^q$, $G(x)= \int_0^x g(t)\,dt$ and $1/p+1/q=1$. A norm is $\lVert f\rVert'_p=\lVert F\rVert_p$. The spaces $L{\!}'^{\,p}$ and $L^p$ are isometrically isomorphic. Distributions in $L{\!}'^{\,p}$ share many properties with functions in $L^p$. Hence, $L{\!}'^{\,p}$ is reflexive, its dual space is identified with $L^q$, there is a type of H\"older inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract $L$-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well-defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes $L{\!}'^{\,1}$ into a Banach algebra isometrically isomorphic to the convolution algebra on $L^1$. Spaces of higher order derivatives of $L^p$ functions are defined. These are also Banach spaces isometrically isomorphic to $L^p$. Archive classification: math.CA math.FA Mathematics Subject Classification: 46E30, 46F10, 46G12 (Primary) 42A38, 42A85, 46B42, 46C05 (Secondary) Submitted from: Erik.Talvila(a)ufv.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.3694 or http://arXiv.org/abs/1208.3694

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Abstract of a paper by Miguel Lacruz
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "Hardy-Littlewood inequalities for norms of positive operators on sequence spaces" by Miguel Lacruz. Abstract: We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator. Archive classification: math.FA Mathematics Subject Classification: 47B37 Remarks: 3 pages, to appear in Lin. Alg. Appl Submitted from: lacruz(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.3246 or http://arXiv.org/abs/1208.3246

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Abstract of a paper by Miguel Lacruz and Maria del Pilar Romero de la Rosa
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "A local spectral condition for strong compactness with some applications to bilateral weighted shifts" by Miguel Lacruz and Maria del Pilar Romero de la Rosa. Abstract: An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly compact} if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fern\'andez-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator. Archive classification: math.FA math.OA Mathematics Subject Classification: 47B07 Remarks: 7 pages, to appear in Proc. Amer. Math. Soc Submitted from: lacruz(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.3245 or http://arXiv.org/abs/1208.3245

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Abstract of a paper by T.Banakh, A.Bartoszewicz, Sz.Glab, and E.Szymonik
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "Algebraic and topological properties of some sets in $l_1$" by T.Banakh, A.Bartoszewicz, Sz.Glab, and E.Szymonik. Abstract: For a sequence $x \in l_1 \setminus c_{00}$, one can consider the set $E(x)$ of all subsums of series $\sum_{n=1}^{\infty} x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets: (I) a finite union of closed intervals; (C) homeomorphic to the Cantor set; (MC) homeomorphic to the set $T$ of subsums of $\sum_{n=1}^\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$. By $I$, $C$ and $MC$ we denote the sets of all sequences $x \in l_1 \setminus c_{00}$, such that $E(x)$ has the corresponding property. In this note we show that $I$ and $C$ are strongly $\mathfrak{c}$-algebrable and $MC$ is $\mathfrak{c}$-lineable. We show that $C$ is a dense $G_\delta$-set in $l_1$ and $I$ is a true $F_\sigma$-set. Finally we show that $I$ is spaceable while $C$ is not spaceable. Archive classification: math.GN math.FA Mathematics Subject Classification: 40A05, 15A03 Remarks: 15 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.3058 or http://arXiv.org/abs/1208.3058

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Abstract of a paper by Christian Rosendal
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "Determinacy of adversarial Gowers games" by Christian Rosendal. Abstract: We prove a game theoretic dichotomy for $G_{\delta\sigma}$ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of $G_{\delta\sigma}$ determinacy. Archive classification: math.LO math.FA Submitted from: rosendal.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.2384 or http://arXiv.org/abs/1208.2384

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Abstract of a paper by E.Ostrovsky and L.Sirota
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "Tchebyshev's characteristic of rearrangement invariant space" by E.Ostrovsky and L.Sirota. Abstract: We introduce and investigate in this short article a new characteristic of rearrangement invariant (r.i.) (symmetric) space, namely so-called Tchebychev's characteristic. We reveal an important class of the r.i. spaces - so called regular r. i. spaces and show that the majority of known r.i. spaces: Lebesgue-Riesz, Grand Lebesgue Spaces, Orlicz, Lorentz and Marcinkiewicz r.i. spaces are regular. But we construct after several examples of r.i. spaces without the regular property. Archive classification: math.FA Submitted from: leos(a)post.sce.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.2393 or http://arXiv.org/abs/1208.2393

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Banach September 2012