## Illinois Journal of Mathematics

### Concerning $q$-summable Szlenk index

Ryan M. Causey

#### Abstract

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.

#### Article information

Source
Illinois J. Math., Volume 62, Number 1-4 (2018), 381-426.

Dates
Revised: 27 November 2018
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.ijm/1552442668

Digital Object Identifier
doi:10.1215/ijm/1552442668

Mathematical Reviews number (MathSciNet)
MR3922422

Zentralblatt MATH identifier
07036792

#### Citation

Causey, Ryan M. Concerning $q$-summable Szlenk index. Illinois J. Math. 62 (2018), no. 1-4, 381--426. doi:10.1215/ijm/1552442668. https://projecteuclid.org/euclid.ijm/1552442668

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