# The Szlenk index of $L_p(X)$ and $A_p$

## TL;DR

This paper establishes optimal relationships between the Szlenk index of a Banach space and its associated $L_p$ spaces, extending previous results to uncountable ordinals and providing new estimates for operators.

## Contribution

It introduces optimal bounds relating the Szlenk index of $X$ and $L_p(X)$, extending prior work to uncountable ordinals and general operators.

## Key findings

- Provides optimal estimates of Szlenk index for $L_p(X)$ in terms of $X$.
- Extends results to uncountable ordinals.
- Estimates Szlenk index of operators $A_p$ in terms of $A$.

## Abstract

Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of H\'ajek and Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.