Arbitrarily distortable Banach spaces of higher order

TL;DR

This paper investigates an ordinal rank on Banach spaces with bases that measures their norm distortion, providing new insights into higher order spreading models and computing this rank for specific classes of Banach spaces.

Contribution

It introduces and analyzes properties of the $AD(\cdot)$ rank, computes it for certain Banach space classes, and answers existing questions about their distortion properties.

Findings

The $AD(\cdot)$ rank is less than $\omega_1$ if and only if the space is arbitrarily distortable.

The rank $AD(\mathfrak{X}^{\omega^\xi}_{0,1})$ equals $\omega^\xi + 1$ for certain classes of Banach spaces.

The paper establishes new results on higher order $\ell_1$ spreading models.

Abstract

We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank $AD(\cdot)$, introduced by P. Dodos, uses the transfinite Schreier familes and has the property that $AD(X) < \omega_1$ if and only if $X$ is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order $\ell_1$ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces $\mathfrak{X}^{\omega^\xi}_{0,1}$ , which each admit $\ell_1$ and $c_0$ spreading models hereditarily, and were introduced by S.A. Argyros, the first and third author, satisfy $AD(\mathfrak{X}^{\omega^\xi}_{0,1}) = \omega^\xi + 1$. This answers some questions of Dodos.