Direct sums and the Szlenk index

TL;DR

This paper characterizes when the Szlenk index of an $ ext{l}_p$-direct sum of operators is bounded by a certain ordinal, linking it to the indices of individual summands, and extends results to $c_0$-direct sums.

Contribution

It provides a necessary and sufficient condition for the Szlenk index of $ ext{l}_p$-direct sums of operators, advancing understanding of their structural properties.

Findings

Szlenk index of $ ext{l}_p$-direct sums is determined by summands' indices

Conditions for Szlenk index bounds are characterized for all $ ext{l}_p$-sums

Results extend to $c_0$-direct sums

Abstract

For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $\omega^\alpha$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $\epsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.