Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Rethinking Polyhedrality for Lindenstrauss Spaces" by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki. Abstract: A recent example by the authors (see arXiv:1503.09088 [math.FA]) shows that an old result of Zippin about the existence of an isometric copy of $c$ in a separable Lindenstrauss space is incorrect. The same example proves that some characterizations of polyhedral Lindenstrauss spaces, based on the result of Zippin, are false. The main result of the present paper provides a new characterization of polyhedrality for the preduals of $\ell_{1}$ and gives a correct proof for one of the older. Indeed, we prove that for a space $X$ such that $X^{*}=\ell_{1}$ the following properties are equivalent: (1) $X$ is a polyhedral space; (2) $X$ does not contain an isometric copy of $c$; (3) $\sup\left\{ x^{*}(x)\,:\, x^{*}\in\mathrm{ext}\left(B_{X^{*}}\right)\setminus D(x)\right\} <1$ for each $x\in S_{X}$, where $D(x)=\left\{ x^{*}\in S_{X^{*}}:x^{*}(x)=1\right\}$. By known theory, from our result follows that a generic Lindenstrauss space is polyhedral if and only if it does not contain an isometric copy of $c$. Moreover, a correct version of the result of Zippin is derived as a corollary of the main result. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 46B25 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/1506.08559 or http://arXiv.org/abs/1506.08559

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Abstract of a paper by Mark Veraar and Lutz Weis
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory" by Mark Veraar and Lutz Weis. Abstract: In this paper we consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form \[L^p(X)\subseteq \gamma(X) \subseteq L^q(X),\] in terms of the type $p$ and cotype $q$ for the Banach space $X$. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein $g$-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions. Archive classification: math.FA Mathematics Subject Classification: Primary 46B09, Secondary: 42B25, 46B70, 46E40, 46B20, 47D07 Submitted from: m.c.veraar(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.08013 or http://arXiv.org/abs/1506.08013

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Abstract of a paper by Stephen J. Dilworth, Denka Kutzarova, Gilles Lancien and Lovasoa N. Randrianarivony
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Equivalent norms with the property $(\beta)$ of Rolewicz" by Stephen J. Dilworth, Denka Kutzarova, Gilles Lancien and Lovasoa N. Randrianarivony. Abstract: We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property $(\beta)$ of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces. Archive classification: math.FA Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.07978 or http://arXiv.org/abs/1506.07978

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Abstract of a paper by Arne Roggensack
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "A short note on the Radon-Riesz property for continuous Banach space valued functions" by Arne Roggensack. Abstract: We present a generalization of the Radon-Riesz property to sequences of continuous functions with values in uniformly convex and uniformly smooth Banach spaces. Archive classification: math.FA Mathematics Subject Classification: Primary: 46B50, Secondary: 46B20 Submitted from: arne.roggensack(a)wias-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.07682 or http://arXiv.org/abs/1506.07682

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Abstract of a paper by Istvan Berkes and Robert Tichy
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "The Kadec-Pe\l czynski theorem in $L^p$, $1\le p<2$" by Istvan Berkes and Robert Tichy. Abstract: By a classical result of Kadec and Pe\l czynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of $\ell^2$ or to the unit vector basis of $\ell^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pe\l czynski theorem is that the limit random measure $\mu$ of the sequence satisfies $\int_{\mathbb{R}} x^2 d\mu (x)\in L^{p/2}$. Archive classification: math.FA Submitted from: berkes(a)tugraz.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.07453 or http://arXiv.org/abs/1506.07453

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Abstract of a paper by Sergei V. Astashkin and Lech Maligranda
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Rademacher functions in Morrey spaces" by Sergei V. Astashkin and Lech Maligranda. Abstract: The Rademacher functions are investigated in the Morrey spaces M(p,w) on [0,1] for 1 \le p <\infty and weight w being a quasi-concave function. They span l_2 space in M(p,w) if and only if the weight w is smaller than the function log_2^{-1/2}(2/t) on (0,1). Moreover, if 1 < p < \infty the Rademacher sunspace R_p is complemented in M(p,w) if and only if it is isomorphic to l_2. However, the Rademacher subspace is not complemented in M(1,w) for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces M(p,w) is described. It turns out that for any infinite-dimensional subspace X of R_p the following alternative holds: either X is isomorphic to l_2 or X contains a subspace which is isomorphic to c_0 and is complemented in R_p. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B42 Remarks: submitted Submitted from: astash(a)samsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.06862 or http://arXiv.org/abs/1506.06862

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Abstract of a paper by E. Dahia, D. Achour, P. Rueda and E. A. Sanchez Perez
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Domination spaces and factorization of linear and multilinear summing operators" by E. Dahia, D. Achour, P. Rueda and E. A. Sanchez Perez. Abstract: It is well known that not every summability property for non linear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. We analyze the class of linear operators that are defined by a summability inequality involving a homogeneous map. Our construction includes the cases of absolutely $p$-summing linear operators, $(p,\sigma)$-absolutely continuous linear operators, factorable strongly $p$-summing multilinear operators, $(p_1,\ldots,p_n)$-dominated multilinear operators and dominated $(p_1,\ldots, p_n;\sigma)$-continuous multilinear operators. Archive classification: math.FA Submitted from: hajdahia(a)univ-msila.dz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.06311 or http://arXiv.org/abs/1506.06311

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Abstract of a paper by Jarno Talponen
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Point-open games and productivity of dense-separable property" by Jarno Talponen. Abstract: In this note we study the point-open topological games to analyze the least upper bound for density of dense subsets of a topological space. This way we may also analyze the behavior of such cardinal invariants in taking products of spaces. Various related cardinal equalities and inequalities are given. As an application we take a look at Banach spaces with the property (CSP) which can be formulated by stating that each weak-star dense linear subspace of the dual is weak-star separable. Archive classification: math.GN math.FA Mathematics Subject Classification: 54A25, 54D70, 91A44, 46B26, 03E60 Submitted from: talponen(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.06080 or http://arXiv.org/abs/1506.06080

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Abstract of a paper by Jean Bourgain
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "On uniformly bounded basis in spaces of holomorphic functions" by Jean Bourgain. Abstract: The main result of the paper is the construction of explicit uniformly bounded basis in the spaces of complex homogenous polynomials on the unit ball of $C^3$, extending an earlier result of the author in the $C^2$ case Archive classification: math.FA Mathematics Subject Classification: Primary: 46E15, 32A99 Secondary: 42A56 Submitted from: bourgain(a)ias.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.05694 or http://arXiv.org/abs/1506.05694

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Abstract of a paper by Assaf Naor
by alspach＠math.okstate.edu 29 Jun '15

29 Jun '15

This is an announcement for the paper "Uniform nonextendability from nets" by Assaf Naor. Abstract: It is shown that there exist Banach spaces $X,Y$, a $1$-net $\mathscr{N}$ of $X$ and a Lipschitz function $f:\mathscr{N}\to Y$ such that every $F:X\to Y$ that extends $f$ is not uniformly continuous. Archive classification: math.MG math.FA Submitted from: naor(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.05391 or http://arXiv.org/abs/1506.05391

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