Szlenk indices of convex hulls

TL;DR

This paper investigates measures of non-compactness in dual Banach spaces, focusing on the Szlenk index, and establishes new relationships and characterizations related to the convex Szlenk index and renorming of Banach spaces.

Contribution

It introduces convexifiable and sublinear measures of non-compactness, proves the equality of Szlenk and convex Szlenk indices for separable spaces, and characterizes spaces with bounded Szlenk index via renorming.

Findings

Szlenk index equals convex Szlenk index in separable spaces.

Characterization of Banach spaces with Szlenk index ≤ ω^{α+1} via renorming.

Extension of previous results on Szlenk index to broader classes of spaces.

Abstract

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their $\omega$-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal $\alpha$, a characterization of the Banach spaces with Szlenk index bounded by $\omega^{\alpha+1}$ in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to $\omega$.