Concerning the Szlenk index

TL;DR

This paper develops combinatorial lemmas on regular families to analyze and compute the Szlenk index of Banach space duals, providing new tools for understanding their structure and constructing spaces with specific Szlenk indices.

Contribution

It introduces pruning and coloring lemmas on regular families and applies them to compute and characterize the Szlenk index in various Banach space contexts.

Findings

Characterization of which countable ordinals occur as Szlenk indices

Computation of Szlenk index of injective tensor products

Optimality of a universality result for Szlenk indices

Abstract

We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove the optimality of a previous universality result, and compute the Szlenk index of the injective tensor product of separable Banach spaces.