The Szlenk Index of L_p(X)

TL;DR

This paper establishes precise bounds relating the Szlenk index and the weak*-dentability index for Banach spaces with separable duals, especially for spaces of the form L_p(X), revealing how these indices compare based on the ordinal type of Sz(X).

Contribution

It provides the first optimal upper bounds on the weak*-dentability index in terms of the Szlenk index for Banach spaces with separable duals, especially for L_p(X) spaces.

Findings

Sz(X) and Dz(X) are tightly bounded for spaces with separable duals.

Explicit bounds depend on whether Sz(X) is a finite or infinite ordinal.

The results clarify the relationship between Szlenk and dentability indices in Banach space theory.

Abstract

We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=\omega^{\alpha}$, for some $\alpha<\omega_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} \omega^{\alpha+1} &\text{if $\alpha$ is a finite ordinal,} \omega^{\alpha} &\text{if $\alpha$ is an infinite ordinal.} {cases}$$