Operator ideals associated with the Szlenk index

TL;DR

This paper studies classes of operators characterized by their Szlenk index, showing that for each ordinal, these classes form well-behaved operator ideals and exploring their relationships with other known ideals.

Contribution

It establishes that the classes of operators with bounded Szlenk index form closed, injective, and surjective operator ideals for all ordinals, and analyzes their connections to existing ideals.

Findings

Each $ ext{SZ}_eta$ class is a closed, injective, surjective operator ideal.

The classes $ ext{SZ}_eta$ relate to several well-known operator ideals.

The structure of $ ext{SZ}_eta$ provides insights into the geometry of Banach spaces.

Abstract

For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$ consisting of all operators whose Szlenk index is an ordinal not exceeding $\omega^\alpha$. Our main result is that $\mathscr{SZ}_\alpha$ is a closed, injective, surjective operator ideal for each $\alpha$. We also study the relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known closed operator ideals.