June 2017 - Banach - mathdept.okstate.edu

Abstract of a paper by Petr Hajek, Thomas Schlumprecht, Andras Zsak
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Generalization of a theorem of Zippin” by Petr Hajek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Andras Zsak<https://arxiv.org/find/math/1/au:+Zsak_A/0/1/0/all/0/1>. Abstract: We generalize and prove a result which was first shown by Zippin, and was explicitly formulated by Benyamini. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10663

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Abstract of a paper by Anna Kamińska, Han Ju Lee, Hyung-Joon Tag
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “M-ideal properties in Orlicz-Lorentz spaces” by Anna Kamińska<https://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1>, Han Ju Lee<https://arxiv.org/find/math/1/au:+Lee_H/0/1/0/all/0/1>, Hyung-Joon Tag<https://arxiv.org/find/math/1/au:+Tag_H/0/1/0/all/0/1>. Abstract: We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $\Lambda_{\phi, w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{\phi, w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to compute the norm of a bounded linear functional on $\Lambda_{\phi, w}$ equipped with Orlicz norm. As a consequence, we show that the order-continuous subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an $M$-ideal in $\Lambda_{\phi, w}$, while this is not true for the space with the Orlicz norm when $\phi$ is an Orlicz $N$-function not satisfying the appropriate $\Delta_2$ condition. The analogous results on Orlicz-Lorentz sequence spaces are given. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10451

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Abstract of a paper by Sheng Zhang
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Asymptotic properties of Banach spaces and coarse quotient maps” by Sheng Zhang<https://arxiv.org/find/math/1/au:+Zhang_S/0/1/0/all/0/1>. Abstract: We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is coarse Lipschitz, then the $(\beta)$-modulus of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10207

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Abstract of a paper by R. M. Causey
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Power type ξ-Asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms” by R. M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $\xi$ -$p$ -asymptotically uniformly smooth norm. We also introduce the notion of $\xi$ -asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $\xi$ -asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $\xi$ -asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the isomorphic properties and isometric properties we study pass from Banach spaces to their injective tensor products. We study the classes of $\xi$ -$p$ -asymptotically uniformly smooth, $\xi$ -$p$ -asymptotically uniformly smoothable, $\xi$ -asymptotically uniformly flat, and $\xi$ -asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.09834

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Abstract of a paper by Florin Catrina, Mikhail I. Ostrovskii
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Images of nowhere differentiable Lipschitz maps of $[0,1]$ into $L_1[0,1]$” by Florin Catrina<https://arxiv.org/find/math/1/au:+Catrina_F/0/1/0/all/0/1>, Mikhail I. Ostrovskii<https://arxiv.org/find/math/1/au:+Ostrovskii_M/0/1/0/all/0/1>. Abstract: The main result: for every $m\in\mathbb{N}$ and $omega>0$ there exists an isometric embedding $F: [0,1]\rightarrow L_1[0,1]$ which is nowhere differentiable, but for each $t\in[0,1]$ the image $F_t$ is an $m$-times continuously differentiable function with absolute values of all of its $m$ derivatives bounded from above by $\omega$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.08916

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Abstract of a paper by Mohammad Sal Moslehian, Ali Zamani, Mahdi Dehghani
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Characterizations of smooth spaces by $\rho_*$-orthogonality” by Mohammad Sal Moslehian<https://arxiv.org/find/math/1/au:+Moslehian_M/0/1/0/all/0/1>, Ali Zamani<https://arxiv.org/find/math/1/au:+Zamani_A/0/1/0/all/0/1>, Mahdi Dehghani<https://arxiv.org/find/math/1/au:+Dehghani_M/0/1/0/all/0/1>. Abstract: The aim of this paper is to present some results concerning the $\rho_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T: X\rightarrow Y$ is a nonzero linear $(I, \rho_*)$-orthogonality preserving mapping between real normed spaces, then $$ 13\|T\|\|x\|\leq \|Tx\|\leq 3\|T\|\|x\|, x\in X $$ where $[T]:=\inf\{\|Tx\|: x\in X, \|x\|=1\}$. We also show that the pair $(X, \perp_{\rho_*})$ is an orthogonality space in the sense of R\"{a}tz. Some characterizations of smooth spaces are given based on the $\rho_*$-orthogonality. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.07032

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Abstract of a paper by Florent Baudier, Gilles Lancien, Thomas Schlumprecht
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Not every infinite dimensional Banach space coarsely contains Hilbert space” by Florent Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>. Abstract: In this article a new concentration inequality is proven for Lipschitz maps on the infinite Hamming graphs and taking values in Tsirelson's original space. This concentration inequality is then used to disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. Some positive embeddability results are proven for the infinite Hamming graphs and the countably branching trees using the theory of spreading models. A purely metric characterization of finite dimensionality is also obtained, as well as a rigidity result pertaining to the spreading model set for Banach spaces coarsely embeddable into Tsirelson's original space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.06797

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Abstract of a paper by
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Power type asymptotically uniformly smooth and asymptotically uniformly flat norms” by Ryan M Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: We provide a short characterization of $p$-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for $1<p<\infty$, the class of $p$-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.05484

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Abstract of a paper by
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper 1-Greedy renormings of Garling sequence spaces” by Fernado Albiac<https://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1>, José L. Ansorena<https://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1>, Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence space provides an example of how essentially non-linear tools from greedy approximation can be used to shed light into the linear structure of the spaces. The paper may be downloaded from t

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Abstract of a paper by Trond A. Abrahamsen, Petr Hájek, Olav Nygaard, Stanimir Troyanski
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper Strongly extreme points and approximation properties” by Trond A. Abrahamsen<https://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1>, Petr Hájek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Olav Nygaard<https://arxiv.org/find/math/1/au:+Nygaard_O/0/1/0/all/0/1>, Stanimir Troyanski<https://arxiv.org/find/math/1/au:+Troyanski_S/0/1/0/all/0/1>. Abstract: We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.02625

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