On strong asymptotic uniform smoothness and convexity

TL;DR

This paper introduces strong asymptotic uniform smoothness and convexity, explores their properties in tensor products and operator spaces, and extends several classical results in Banach space theory.

Contribution

It defines new notions of strong asymptotic uniform smoothness and convexity, and applies these to tensor products, operator spaces, and characterizes when spaces of compact operators are smooth.

Findings

Injective tensor product of strongly asymptotically uniformly smooth spaces is smooth.

Characterization of Orlicz functions for which operator spaces are smooth.

Spaces of compact operators are not strictly convex in higher dimensions.

Abstract

We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our techniques also provide a characterisation of Orlicz functions $M, N$ such that the space of compact operators $\mathcal K(h_M,h_N)$ is asymptotically uniformly smooth. Finally we show that $\mathcal K(X, Y)$ is not strictly convex whenever $X$ and $Y$ are at least two-dimensional, which extends a result by Dilworth and Kutzarova.