Banach June 2017

banach@mathdept.okstate.edu

1 participants
14 discussions

Abstract of a paper by Luis García-Lirola, Matías Raja
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “On strong asymptotic uniform smoothness and convexity” by Luis García-Lirola<https://arxiv.org/find/math/1/au:+Garcia_Lirola_L/0/1/0/all/0/1>, Matías Raja<https://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1>. Abstract: We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our techniques also provide a characterisation of Orlicz functions $M, N$ such that the space of compact operators $\mathcal{K}(h_M, h_N)$ is asymptotically uniformly smooth. Finally we show that $\mathcal{K}(X, Y)$ is not strictly convex whenever $X$ and $Y$ are at least two-dimensional, which extends a result by Dilworth and Kutzarova. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.02020

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

01 Jun '17

This is an announcement for the paper “On the Normality, Regularity and Chain-completeness of Partially Ordered Banach Spaces and Applications” by Jinlu Li<https://arxiv.org/find/math/1/au:+Li_J/0/1/0/all/0/1>. Abstract: In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially ordered Banach spaces. As applications of these fixed point theorems, we prove the existence of solutions of some integral equations, such as Hammerstein integral equations, in Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.01580

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Abstract of a paper by Petr Hájek, Tommaso Russo
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Some remarks on smooth renormings of Banach spaces” by Petr Hájek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Tommaso Russo<https://arxiv.org/find/math/1/au:+Russo_T/0/1/0/all/0/1>. Abstract: We prove that in every separable Banach space X with a Schauder basis and a $C_k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C_k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.01384

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Abstract of a paper by Gilles Lancien, Matias Raja
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces” by Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>, Matias Raja<https://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1>. Abstract: In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\leq q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.00577

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Banach June 2017