Vector-valued numerical radius and $\sigma$-porosity

Abstract

It is well known that under certain conditions on a Banach space $X$, the set of bounded linear operators attaining their numerical radius is a dense subset. We prove in this paper that if $X$ is assumed to be uniformly convex and uniformly smooth then the set of bounded linear operators attaining their numerical radius is not only a dense subset but also the complement of a $\sigma$-porous subset. In fact, we generalize the notion of numerical radius to a large class $\mathcal{Z}$ of vector-valued operators defined from $X\times X^{\ast }$ into a Banach space $W$ and we prove that the set of all elements of $\mathcal{Z}$ strongly (up to a symmetry) attaining their {\it numerical radius} is the complement of a $\sigma$-porous subset of $\mathcal{Z}$ and moreover the {\it "numerical radius"} {\it Bishop-Phelps-Bollobás property} is also satisfied for this class. Our results extend (up to the assumption on $X$) some known results in several directions: $(1)$ the density is replaced by being the complement of a $\sigma$-porous subset, $(2)$ the operators attaining their {\it numerical radius} are replaced by operators strongly (up to a symmetry) attaining their {\it numerical radius} and $(3)$ the results are obtained in the vector-valued framework for general linear and non-linear vector-valued operators (including bilinear mappings and the classical space of bounded linear operators).