Prescribed Szlenk index of separable Banch spaces

TL;DR

This paper characterizes the possible Szlenk indices of separable Banach spaces, constructing spaces with prescribed indices and exploring their properties, including reflexivity and basis structures.

Contribution

It completes the classification of Szlenk indices for separable Banach spaces by constructing spaces with specific index values and analyzing their properties.

Findings

Existence of spaces with prescribed Szlenk indices for duals of various orders.

Lindenstrauss space and its dual have Szlenk index ω.

Any element of the set can be realized as a Szlenk index of a reflexive space with an unconditional basis.

Abstract

In a previous work, the first named author described the set $\cal P$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\cal P$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in $\{0,..,n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$. We also show that any element of $\cal P$ can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.