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2018 Concerning $q$-summable Szlenk index
Ryan M. Causey
Illinois J. Math. 62(1-4): 381-426 (2018). DOI: 10.1215/ijm/1552442668


For each ordinal $\xi$ and each $1\leqslant q<\infty$, 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 $\alpha_{\xi,p}$ 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 $\alpha_{\xi,p}$ 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 $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, 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 $\alpha_{\xi,p}$ 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.


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Ryan M. Causey. "Concerning $q$-summable Szlenk index." Illinois J. Math. 62 (1-4) 381 - 426, 2018.


Received: 27 November 2018; Revised: 27 November 2018; Published: 2018
First available in Project Euclid: 13 March 2019

zbMATH: 07036792
MathSciNet: MR3922422
Digital Object Identifier: 10.1215/ijm/1552442668

Primary: 46B03, 46B06
Secondary: 46B28, 47B10

Rights: Copyright © 2018 University of Illinois at Urbana-Champaign


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Vol.62 • No. 1-4 • 2018
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