Arbitrarily distortable Banach spaces of higher order

2014/08/21 by Beanland, Kevin, Causey, Ryan, Motakis, Pavlos
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1408.5065

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(⋅), introduced by P. Dodos, uses the transfinite Schreier familes and has the property that AD(X) < ω₁ if and only if X is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order ℓ₁ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces \mathfrakX^ωξ_0,1 , which each admit ℓ₁ and c₀ spreading models hereditarily, and were introduced by S.A. Argyros, the first and third author, satisfy AD(\mathfrakX^ωξ_0,1) = ω^ξ+ 1. This answers some questions of Dodos.

Related

• The Distorion Problem
• The distortion problem
• A remark about distortion
• Rank-one perturbations and norm-attaining operators
• Note on distortion and Bourgain ℓ₁ index