August 2016 - Banach - mathdept.okstate.edu

Abstract of a paper by Dongyang Chen, J. Alejandro Chávez-Domínguez, Lei Li
by Bentuo Zheng (bzheng) 31 Jul '16

31 Jul '16

This is an announcement for the paper “Unconditionally p-converging operators and Dunford-Pettis Property of order $p$” by Dongyang Chen<http://arxiv.org/find/math/1/au:+Chen_D/0/1/0/all/0/1>, J. Alejandro Chávez-Domínguez<http://arxiv.org/find/math/1/au:+Ch%7Ba%7Dvez_Dom%7Bi%7Dnguez_J/0/1/0/all/0…>, Lei Li<http://arxiv.org/find/math/1/au:+Li_L/0/1/0/all/0/1>. Abstract: In the present paper we study unconditionally $p$-converging operators and Dunford-Pettis property of order $p$. New characterizations of unconditionally $p$-converging operators and Dunford-Pettis property of order $p$ are established. Six quantities are defined to measure how far an operator is from being unconditionally $p$-converging. We prove quantitative versions of relationships of completely continuous operators,unconditionally $p$-converging operators and unconditionally converging operators. We further investigate possible quantifications of the Dunford-Pettis property of order $p$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.02161

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Abstract of a paper by Richard M. Aron, Jesús A. Jaramillo, Enrico Le Donne
by Bentuo Zheng (bzheng) 31 Jul '16

31 Jul '16

This is an announcement for the paper “Smooth surjections and surjective restrictions” by Richard M. Aron<http://arxiv.org/find/math/1/au:+Aron_R/0/1/0/all/0/1>, Jesús A. Jaramillo<http://arxiv.org/find/math/1/au:+Jaramillo_J/0/1/0/all/0/1>, Enrico Le Donne<http://arxiv.org/find/math/1/au:+Donne_E/0/1/0/all/0/1>. Abstract: Given a surjective mapping $f: E\rightarrow F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C_1$ -smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $R_n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01725

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Abstract of a paper by Anton R Schep
by Bentuo Zheng (bzheng) 31 Jul '16

31 Jul '16

This is an announcement for the paper “Unbounded Disjointness Preserving Linear Functionals and Operators” by Anton R Schep<http://arxiv.org/find/math/1/au:+Schep_A/0/1/0/all/0/1>. Abstract: Let $E$ and $F$ be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice $E$, which shows that in this case the unbounded disjointness operators from $E\rightarrow F$ separate the points of $E$. Then we show that every disjointness preserving operator $T: E\rightarrow F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T: E\rightarrow F$ has a unique decomposition $T=R+S$, where $R$ is a bounded disjointness preserving operator and $S$ is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that $E=C(X)$ with $X$ a compact Hausdorff space, we show that every disjointness preserving operator $T: C(X)\rightarrow F$ is norm bounded on an norm dense sublattice algebra of $C(X)$, which leads then to a decomposition of $T$ into a bounded disjointness operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01423

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Abstract of a paper by Ryan M. Causey, Stephen J. Dilworth
by Bentuo Zheng (bzheng) 31 Jul '16

31 Jul '16

This is an announcement for the paper “$\xi$-asymptotically uniformly smooth, $\xi$--asymptotically uniformly convex, and $(\beta)$-operators” by Ryan M. Causey<http://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, Stephen J. Dilworth<http://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$, we define the notions of $\xi$-asymptotically uniformly smooth and $w^*$-$\xi$-asymptotically uniformly convex operators. When $\xi=0$, these extend the notions of asymptotically uniformly smooth and $w^*$-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property $(\beta)$ of Rolewicz which extends the notion of property $(\beta)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(\beta)$ in terms of the Szlenk index of the operator and its adjoint. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01362

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