Prescribed Szlenk index of separable Banch spaces

Abstract

In a previous work, the first named author described the set $\mathcal{P}$ 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 $\mathcal{P}$, 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 $\mathcal{P}$ can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.

17 pages. It is a revised version of the previous preprint "Prescribed Szlenk index of iterated duals": arXiv:1710.01638. The paper has been reorganized and the title has been changed. To appear in Studia Math

References in corpus (1)

• Direct sums and the Szlenk index