Asplund operators and the Szlenk index

TL;DR

This paper studies classes of operators characterized by their Szlenk index, showing they form closed operator ideals and exploring their relationships with known ideals, with applications to Asplund operators.

Contribution

It introduces and analyzes the classes  ext{SZ}_ ext{_ ext{alpha}} of operators with bounded Szlenk index, establishing their ideal properties and connections to other operator classes.

Findings

Each  ext{SZ}_ ext{_ ext{alpha}} is a closed operator ideal.

Relationships between  ext{SZ}_ ext{_ ext{alpha}} and known ideals are characterized.

Quantitative factorization results for Asplund operators are derived.

Abstract

For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$ consisting of all operators whose Szlenk index is an ordinal not exceeding $\omega^\alpha$. We show that each class $\mathscr{SZ}_\alpha$ is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known closed operator ideals is investigated and quantitative factorization results in terms of the Szlenk index are obtained for the class of Asplund operators.