Absolutely convex sets of large Szlenk index

TL;DR

This paper investigates the properties of Banach spaces and operators with large or undefined Szlenk index, establishing subspace and quotient structures with comparable Szlenk indices and characterizing universal operators.

Contribution

It introduces new structural results for Banach spaces with large Szlenk index and characterizes the existence of universal operators based on Szlenk index bounds.

Findings

Existence of subspaces with basis and comparable Szlenk indices

Existence of quotients with basis and comparable Szlenk indices

Characterization of universal operators based on Szlenk index bounds

Abstract

Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. We study consequences of $K$ having a large or undefined Szlenk index and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the Szlenk indices of $X$. We also show that for a given ordinal $\xi$ the class of operators whose Szlenk index is not an ordinal less than or equal to $\xi$ admits a universal element if and only if $\xi<\omega_1$; W.B. Johnson's theorem that the formal identity map from $\ell_1$ to $\ell_\infty$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.