Arbitrarily distortable Banach spaces of higher order

Kevin Beanland; Ryan Causey; Pavlos Motakis

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arXiv · 1408.5065

Arbitrarily distortable Banach spaces of higher order

Kevin Beanland·Ryan Causey·Pavlos Motakis

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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.

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BibTeX RIS

Kevin Beanland, Ryan Causey, Pavlos Motakis. 2014-08-21. Arbitrarily distortable Banach spaces of higher order. https://arxiv.org/abs/1408.5065

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