Banach

banach@mathdept.okstate.edu

2549 discussions

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

31 Jul '16

This is an announcement for the paper “Pełczyński's property $(V^*)$ of order $p$ and its quantification” by Lei Li<http://arxiv.org/find/math/1/au:+Li_L/0/1/0/all/0/1>, 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…>. Abstract: We introduce the concepts of Pe{\l}czy\'{n}ski's property $(V)$ of order $p$ and Pe{\l}czy\'{n}ski's property $(V^*)$ of order $p$. It is proved that, for each $1<p<\infty$, the James $p$-space $J_p$ enjoys Pe{\l}czy\'{n}ski's property $(V^*)$ of order $p$ and the James $p^*$-space $J_{p^*}$ (where $p^*$ denotes the conjugate number of $p$) enjoys Pe{\l}czy\'{n}ski's property $(V)$ of order $p$. We prove that both $L_1(\mu)$ ($\mu$ a finite positive measure) and $\ell_1$ enjoy the quantitative version of Pe{\l}czy\'{n}ski's property $(V^*)$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.02163

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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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for posting in the BBB
by GODEFROY Gilles 27 Jul '16

27 Jul '16

We are glad to announce the publication of the two volumes of the Kalton Selecta by Birkhauser, in the series ``Contemporary Mathematicians ». These two volumes (of about 780 pages each) gather 43 articles written by Nigel Kalton (1946-2010), sometimes with co-authors, with comments by experts of the many fields to which Nigel contributed, namely: F. Albiac, J. L. Ansorena, P. H. Enflo, D. H. Fremlin, A. Kaminska, B. Maurey, R. J. Elliott, F. Gesztesy, T. Kucherenko, G. Lancien, V. Peller, P. Portal, F. Sukochev, S. A. Ivanov, N. K. Nikolskii, I. E. Verbitsky, L. Weis, Y. A. Brudnyi, S. J. Dilworth, D. Kutzarova, Y. Benyamini, J. M. F. Castillo, H. G. Dales, M. I. Ostrovskii, G. Pisier, D. Werner, M. Cwikel, M. Milman, R. Rochberg, S. J.Dilworth, S. J. Montgomery-Smith. These Kalton Selecta are not only a tribute to an outstanding mathematician and his formidable contribution to functional analysis. They constitute a work in which every article is explained and put in its proper framework, in such a way that the reader gets easy access to advanced theories, through the efforts of the contributors. Their appearance is a scientific event for experts and beginners alike. Nigel Kalton’s work is a gold mine for anyone who is ready to read his lines - and in between. F. Gesztesy, G. Godefroy, L. Grafakos, I. Verbitsky.

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Abstract of a paper by Yisheng Song, Rudong Chen
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Pazy's fixed point theorem with respect to the partial order in uniformly convex Banach spaces” by Yisheng Song<http://arxiv.org/find/math/1/au:+Song_Y/0/1/0/all/0/1>, Rudong Chen<http://arxiv.org/find/math/1/au:+Chen_R/0/1/0/all/0/1>. Abstract: In this paper, the Pazy's Fixed Point Theorems of monotone $\alpha$−nonexpansive mapping $T$ are proved in a uniformly convex Banach space $E$ with the partial order "$\leq$". That is, we obtain that the fixed point set of $T$ with respect to the partial order "$\leq$" is nonempty whenever the Picard iteration $\{T^n x_0\}$ is bounded for some initial point $x_0$ with $x_0\leq Tx_0$ or $Tx_0\leq x_0$. When restricting the demain of $T$ to the cone $P$, a monotone $\alpha$−nonexpansive mapping $T$ has at least a fixed point if and only if the Picard iteration $\{T^n0\}$ is bounbed. Furthermore, with the help of the properties of the normal cone $P$, the weakly and strongly convergent theorems of the Picard iteration $\{T^n x_0\}$ are showed for finding a fixed point of $T$ with respect to the partial order "$\leq$" in uniformly convex ordered Banach space. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.08216

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Abstract of a paper by Petr Hájek, Matěj Novotný
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Some remarks on the structure of Lipschitz-free spaces” by Petr Hájek<http://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Matěj Novotný<http://arxiv.org/find/math/1/au:+Novotny_M/0/1/0/all/0/1>. Abstract: We give several structural results concerning the Lipschitz-free spaces $F(M)$ where $M$ is a metric space. We show that $F(M)$ contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=$dens$(M)$. If $\mathcal{N}$ is the net in a finite dimensional Banach space $X$, we show that $F(\mathcal{N})$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $F(\mathcal{N})$ is isomorphic to its $\ell_1$-sum. Finally, we prove that for all $X\equiv C(K)$ spaces $F(\mathcal{N})$ are mutually isomorphic spaces with a Schauder basis. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.03926

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Abstract of a paper by C. Ward Henson, Yves Raynaud
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Asymptotically Hilbertian Modular Banach Spaces: Examples of Uncountable Categoricity” by C. Ward Henson<http://arxiv.org/find/math/1/au:+Henson_C/0/1/0/all/0/1>, Yves Raynaud<http://arxiv.org/find/math/1/au:+Raynaud_Y/0/1/0/all/0/1>. Abstract: We give a criterion ensuring that the elementary class of a modular Banach space $E$ (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of $E$) consists of all direct sums $E\oplus_m H$, where $H$ is an arbitrary Hilbert space and $\oplus_m$ denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.03122

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Abstract of a paper by Amra Rekic-Vukovic, Nermin Okicic, Ivan Arandjelovic
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “On modulus of noncompact convexity for a strictly minimalizable measure of noncompactness” by Amra Rekic-Vukovic<http://arxiv.org/find/math/1/au:+Rekic_Vukovic_A/0/1/0/all/0/1>, Nermin Okicic<http://arxiv.org/find/math/1/au:+Okicic_N/0/1/0/all/0/1>, Ivan Arandjelovic<http://arxiv.org/find/math/1/au:+Arandjelovic_I/0/1/0/all/0/1>. Abstract: In this paper we consider modulus of noncompact convexity $\Delta_{X, \Phi}$ associated with the strictly minimalizable measure of noncompactness $\Phi$. We also give some its properties and show its continuity on the interval $[0, \Phi(\bar{B}_X))$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.01071

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Banach