Rough norms in spaces of operators

TL;DR

This paper characterizes when spaces of bounded linear operators are rough or average rough, linking properties of the domain and codomain spaces to the operator space's geometric roughness.

Contribution

It provides necessary and sufficient conditions for operator spaces to be rough or average rough, unifying and improving previous theorems in the field.

Findings

Operator space is $ ext{delta}$-average rough if the dual space is $ ext{delta}$-average rough and the codomain is alternatively octahedral.

Unified framework improves previous results by Becerra Guerrero et al.

Establishes new connections between geometric properties of Banach spaces and operator spaces.

Abstract

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that $\mathcal L(X,Y)$ is $\delta$-average rough whenever $X^\ast$ is $\delta$-average rough and $Y$ is alternatively octahedral. This allows us to give a unified improvement of two theorems by Becerra Guerrero, L\'opez-P\'erez, and Rueda Zoca [J. Math. Anal. Appl. 427 (2015)].