Direct sums and summability of the Szlenk index

TL;DR

This paper investigates how the Szlenk index behaves under various direct sum constructions of Banach spaces, establishing conditions for summability and deriving formulas for power types.

Contribution

It proves the preservation of summable Szlenk index under $c_0$-sums and provides a formula for Szlenk power type in certain $rak{E}$-direct sums, extending understanding of Banach space structures.

Findings

The $c_0$-sum of spaces with uniformly summable Szlenk index also has summable Szlenk index.

The Szlenk power type of certain $rak{E}$-direct sums can be explicitly calculated.

The Tsirelson direct sum of infinitely many $c_0$ spaces has power type 1 but non-summable Szlenk index.

Abstract

We prove that the $c_0$-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the $\mathfrak{E}$-direct sum of separable spaces provided that $\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\ell_p$ structure in $\mathfrak{E}^\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.