June 2018 - Banach - mathdept.okstate.edu

Abstract of a paper by R.M. Causey, K. Navoyan
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Factorization of Asplund operators” by R.M. Causey<https://arxiv.org/search?searchtype=author&query=Causey%2C+R+M>, K. Navoyan<https://arxiv.org/search?searchtype=author&query=Navoyan%2C+K>. Abstract: We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $\xi\in (0, \omega_1)\setminus\{\omega^\eta: \eta<\omega_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_\xi$ having a shrinking basis and Szlenk index $\omega^\xi$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $\omega^\xi$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_\xi$, and if $X$ has a shrinking FDD, $A$ factors through $\mathfrak{G}_\xi$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.02746

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Abstract of a paper by Fernando Albiac, José L. Ansorena, Pablo M. Berná
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Asymptotic greediness of the Haar system in the spaces $L_p[0,1], 1<p<\infty$” by Fernando Albiac<https://arxiv.org/search?searchtype=author&query=Albiac%2C+F>, José L. Ansorena<https://arxiv.org/search?searchtype=author&query=Ansorena%2C+J+L>, Pablo M. Berná<https://arxiv.org/search?searchtype=author&query=Bern%C3%A1%2C+P+M>. Abstract: Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max\{p,p/(p-1)\}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that $p^{\ast}\lesssim C_g[\mathcal{H}^{(p)},L_p]\lesssim (p^{\ast})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_g[\mathcal{H}^{(p)},L_p]\approx p^{\ast}$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.01528

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Abstract of a paper by Sergey V. Astashkin
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Duality problem for disjointly homogeneous rearrangement invariant spaces” by Sergey V. Astashkin<https://arxiv.org/search?searchtype=author&query=Astashkin%2C+S+V>. Abstract: Let $1\leq p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $\ell_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1< p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $\ell_p$, $1\leq p<\infty$, or $c_0$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.00691

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