Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

TL;DR

This paper constructs specific reflexive Banach spaces with a basis that have a precise Szlenk index, serving as universal models for classes of spaces with bounded Szlenk indices, advancing understanding of Banach space complexity.

Contribution

It introduces a method to estimate the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces, providing new universal examples for classes defined by Szlenk index bounds.

Findings

Existence of separable, reflexive Banach spaces with prescribed Szlenk index.

Construction of universal spaces for classes with bounded Szlenk indices.

Extension of the theory of Szlenk index using generalized Baernstein spaces.

Abstract

For each ordinal $\alpha< \omega_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $\omega^\alpha$.