Power type $\xi$-asymptotically uniformly smooth norms

TL;DR

This paper generalizes the theory of asymptotically uniformly smooth norms in Banach spaces by introducing $\xi$-Szlenk power type, characterizing operators and spaces with such norms, and analyzing their ideal and factorization properties.

Contribution

It extends existing renorming results to all ordinals $\xi$, providing a detailed characterization and introducing the $\xi$-Szlenk power type for Banach space analysis.

Findings

Characterized operators admitting $\xi$-asymptotically uniformly smooth norms.

Computed the optimal exponent for the $\xi$-Szlenk power type.

Analyzed ideal and factorization properties of classes with $\xi$-Szlenk power type.

Abstract

We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $\omega$. For every ordinal $\xi$, we characterize the operators, and therefore the Banach spaces, which admit a $\xi$-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $Sz_\xi(\cdot, \varepsilon)$. We also introduce the $\xi$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $\xi$-Szlenk power type.