The Szlenk index of injective tensor products and convex hulls

TL;DR

This paper calculates the Szlenk index for convex hulls and injective tensor products of Banach spaces, providing explicit formulas and characterizations for these indices and their relation to space properties.

Contribution

It introduces formulas for Szlenk indices of convex hulls and tensor products, and characterizes possible Szlenk and Bourgain indices for Banach spaces.

Findings

Szlenk index of convex hulls expressed via original set

Szlenk index of injective tensor products in terms of component spaces

Complete characterization of possible Szlenk and Bourgain indices

Abstract

Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ index of a Banach space.