Vector-valued numerical radius and σ-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 σ-porous subset. In fact, we generalize the notion of numerical radius to a large class Z of vector-valued operators defined from X×X⁎ into a Banach space W and we prove that the set of all elements of Z strongly (up to a symmetry) attaining their numerical radius is the complement of a σ-porous subset of Z and moreover the “numerical radius” 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 σ-porous subset, (2) the operators attaining their numerical radius are replaced by operators strongly (up to a symmetry) attaining their 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).