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 -nu and L -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 but also on a state or on a numerical radius one operator T. We address the relation between the L -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and are examples of Banach spaces satisfying the L -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 -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.