February 2018 - Banach - mathdept.okstate.edu

Abstract of a paper by Francisco Javier Garcia-Pacheco, Francisco Javier Perez-Fernandez
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Vector-Valued Banach Limits and Vector-Valued Almost Convergence” by Francisco Javier Garcia-Pacheco<https://arxiv.org/find/math/1/au:+Garcia_Pacheco_F/0/1/0/all/0/1>, Francisco Javier Perez-Fernandez<https://arxiv.org/find/math/1/au:+Perez_Fernandez_F/0/1/0/all/0/1>. Abstract: This book pretends to compile the latest advances on vector-valued Banach limits as well as their applications to vector-valued almost convergence. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.06235

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Abstract of a paper by Eftychios Glakousakis, Sophocles Mercourakis
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Antipodal sets in infinite dimensional Banach spaces” by Eftychios Glakousakis<https://arxiv.org/find/math/1/au:+Glakousakis_E/0/1/0/all/0/1>, Sophocles Mercourakis<https://arxiv.org/find/math/1/au:+Mercourakis_S/0/1/0/all/0/1>. Abstract: The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)$−separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subset S_X$ and a constant$d>1$, satisfying the property that for every $x, y\in S$ with $x\neq y$ there exists $f\in B_{X^*}$ such that $d\leq f(x)-f(y)$ and $f(y)\leq f(z)\leq f(x)$, for all $z\in S$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.02002

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Abstract of a paper by Saak Gabriyelyan
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Locally convex spaces and Schur type properties” by Saak Gabriyelyan<https://arxiv.org/find/math/1/au:+Gabriyelyan_S/0/1/0/all/0/1>. Abstract: We extend Rosenthal's characterization of Banach spaces with the Schur property to a wide class of locally convex spaces (lcs) strictly containing the class of Fr\'{e}chet spaces by showing that for an lcs $E$ from this class the following conditions are equivalent: (1) $E$ has the Schur property, (2) $E$ and $E_w$ have the same sequentially compact sets, where $E_w$ is the space $E$ with the weak topology, (3) $E$ and $E_w$ have the same compact sets, (4) $E$ and $E_w$ have the same countably compact sets, (5) $E$ and $E_w$ have the same pseudocompact sets, (6) $E$ and $E_w$ have the same functionally bounded sets, (7) every bounded non-precompact sequence in $E$ has a subsequence which is equivalent to the unit basis of $\ell_1$. We show that for a quasi-complete lcs conditions (3)-(6) are equivalent to (8) every non-precompact bounded subset of $E$ has an infinite subset which is discrete and $C$-embedded in $E_w$. We prove that every real locally convex space is a quotient space of an lcs $E$ satisfying conditions (1)-(5). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.01992

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Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Concerning $q$-summable Szlenk index” by Ryan M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$ and each $1\leq q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $a_{\xi, p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $a_{\xi, p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $a_{\xi, p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $a_{\xi, p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $a_{\xi, p}$ seminorms under $\ell_r$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_p$ and $c_0$ direct sums of operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.00033

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