Concerning $q$-summable Szlenk index

TL;DR

This paper introduces a generalized notion of $\xi$-$q$-summable Szlenk index for Banach spaces, explores its properties, and demonstrates how it influences the structure and embeddings of Banach spaces with respect to tensor products, bases, and direct sums.

Contribution

It defines the $\xi$-$q$-summable Szlenk index, relates it to the $ au_{\xi,p}$ seminorm, and studies its behavior under tensor products, embeddings, and direct sums, extending previous Szlenk index results.

Findings

The $ au_{\xi,p}$ seminorm is determined by norming sets.

Behavior under tensor products is characterized.

Embeddings preserve $ au_{\xi,p}$ properties and Szlenk power type.

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.