October 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by Masato Mimura
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Sphere equivalence, Banach expanders, and extrapolation" by Masato Mimura. Abstract: We study the Banach spectral gap lambda_1(G;X,p) of finite graphs G for pairs (X,p) of Banach spaces and exponents. We introduce the notion of sphere equivalence between Banach spaces, and study behavior of lambda_1(G;X,p) for fixed p in terms of this equivalence. We further study behavior of lambda_1(G;X,p) for fixed X. As a byproduct, we show a generalization of Matousek's extrapolation to that for any Banach space which is sphere equivalent to a uniformly convex Banach space. We as well prove that expanders are expanders with respects to (X,p) for any X sphere equivalent to a uniformly curved Banach space and for any finite p strictly bigger than 1. Archive classification: math.GR math.CO math.FA math.MG Remarks: 23 pages, no figure Submitted from: mimura-mas(a)m.tohoku.ac.jp The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4737 or http://arXiv.org/abs/1310.4737

1 0

Abstract of a paper by Anna Novikova
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Lyapunov theorem for q-concave Banach spaces" by Anna Novikova. Abstract: Generalization of Lyapunov convexity theorem is proved for vector measure with values in Banach spaces with unconditional bases, which are q-concave for some $q<\infty.$ Archive classification: math.FA Mathematics Subject Classification: 46E30 Remarks: 7 pages Submitted from: anna.novikova(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4663 or http://arXiv.org/abs/1310.4663

1 0

Abstract of a paper by Niushan Gao
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Unbounded order convergence in dual spaces" by Niushan Gao. Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to $0$ in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $\sigma$-order complete Banach lattices the spaces in whose dual space every simultaneously uo- and $w^*$-convergent sequence converges weakly/in norm. Archive classification: math.FA Submitted from: niushan(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4438 or http://arXiv.org/abs/1310.4438

1 0

Abstract of a paper by Paul F.X. Muller and Johanna Penteker
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "p-summing multiplication operators, dyadic Hardy Spaces and atomic decomposition" by Paul F.X. Muller and Johanna Penteker. Abstract: We constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (\varphi_I) \mapsto \sum \varphi_Ix_Ih_I. \] Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition. Archive classification: math.FA Mathematics Subject Classification: 42B30 46B25 46B09 46B42 46E40 47B10 60G42 Remarks: 24 pages Submitted from: johanna.penteker(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4312 or http://arXiv.org/abs/1310.4312

1 0

Abstract of a paper by M. A. Sofi
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Some problems in functional analysis inspired by Hahn Banach type theorems" by M. A. Sofi. Abstract: As a cornerstone of functional analysis, Hahn Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem. Archive classification: math.FA Mathematics Subject Classification: 46B20, 47B10, 46G10 Remarks: 29 pages, 0 figures, accepted in Ann. Func. Anal Submitted from: aminsofi(a)kashmiruniversity.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.3382 or http://arXiv.org/abs/1310.3382

1 0

Abstract of a paper by Tomasz Kania and Richard J. Smith
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "A note on the Bishop property in compact spaces" by Tomasz Kania and Richard J. Smith. Abstract: We answer two questions concerning the Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author. There are two versions of ($\symbishop$): one applies to linear operators and the other to compact Hausdorff spaces. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented. Archive classification: math.GN math.FA 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/1310.4035 or http://arXiv.org/abs/1310.4035

1 0

Abstract of a paper by Joanna Garbulinska-Wegrzyn and Wieslaw Kubis
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "A universal operator on the Gurarii space" by Joanna Garbulinska-Wegrzyn and Wieslaw Kubis. Abstract: We construct a nonexpansive linear operator on the Gurarii space that ``captures" all nonexpansive linear operators between separable Banach spaces. Some additional properties involving its restrictions to finite-dimensional subspaces describe this operator uniquely up to an isometry. Archive classification: math.FA Mathematics Subject Classification: 47A05, 47A65, 46B04 Remarks: 17 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.2380 or http://arXiv.org/abs/1310.2380

1 0

Abstract of a paper by Gideon Schechtman
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "No greedy bases for matrix spaces with mixed $\ell_p$ and $\ell_q$" by Gideon Schechtman. Abstract: We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, $1\le p<\infty$, and $(\bigoplus_{n=1}^\infty c_o)_{\ell_q}$, $1\le q<\infty$, do not have greedy bases. It follows from that and known results that a class of Besov spaces on $\R^n$ lack greedy bases as well. Archive classification: math.FA Mathematics Subject Classification: 46B15, 41A65, 46B45, 46E35 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.2371 or http://arXiv.org/abs/1310.2371

1 0

Abstract of a paper by Claudia Correa and Daniel V. Tausk
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "Compact lines and the Sobczyk property" by Claudia Correa and Daniel V. Tausk. Abstract: We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46E15, 54F05 Remarks: 12 pages Submitted from: tausk(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.1950 or http://arXiv.org/abs/1310.1950

1 0

Abstract of a paper by Foivos Xanthos
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "A version of Kalton's theorem for the space of regular operators" by Foivos Xanthos. Abstract: In this note we extend some recent results in the space of regular operators. In particular, we provide the following Banach lattice version of a classical result of Kalton: Let $E$ be an atomic Banach lattice with an order continuous norm and $F$ a Banach lattice. Then the following are equivalent: (i) $L^r(E,F)$ contains no copy of $\ell_\infty$, \,\, (ii) $L^r(E,F)$ contains no copy of $c_0$, \,\, (iii) $K^r(E,F)$ contains no copy of $c_0$, \,\, (iv) $K^r(E,F)$ is a (projection) band in $L^r(E,F)$, \,\, (v) $K^r(E,F)=L^r(E,F)$. Archive classification: math.FA Submitted from: foivos(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.1591 or http://arXiv.org/abs/1310.1591

1 0