$\xi$-asymptotically uniformly smooth, $\xi$-asymptotically uniformly convex, and $(\beta)$ operators

TL;DR

This paper introduces and characterizes $\xi$-asymptotically uniformly smooth and convex operators, extending classical notions to operators via Szlenk index, and explores their duality and renorming properties, including property $(eta)$.

Contribution

It defines new operator classes based on asymptotic smoothness and convexity, relates them to Szlenk index, and extends property $(eta)$ to operators with a comprehensive duality and renorming framework.

Findings

Complete description of renorming conditions via Szlenk index.

Duality characterization between $\xi$-asymptotic smoothness and convexity.

Extension of property $(eta)$ to operators with renorming criteria.

Abstract

For each ordinal $\xi$, we define the notions of $\xi$-asymptotically uniformly smooth and $w^*$-$\xi$-asymptotically uniformly convex operators. When $\xi=0$, these extend the notions of asymptotically uniformly smooth and $w^*$-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property $(\beta)$ of Rolewicz which extends the notion of property $(\beta)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(\beta)$ in terms of the Szlenk index of the operator and its adjoint.