On various types of density of numerical radius attaining operators

Abstract

In this paper, we are interested in studying Bishop-Phelps-Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop-Phelps-Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop-Phelps-Bollobás point property. We also consider two weaker properties L p,p -nu and L o,o -nu, the local versions of Bishop-Phelps-Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on ε > 0 but also on a state (x, x * ) or on a numerical radius one operator T . We address the relation between the L p,p -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and c 0 are examples of Banach spaces satisfying the L p,p -nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L o,o -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.