On various types of density of numerical radius attaining operators

Abstract

In this paper, we are interested in studying Bishop-Phelps-Bollobas 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-Bollobas point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop-Phelps-Bollobas point property. We also consider two weaker properties L-p,L-p-nu and L-o,L-o-nu, the local versions of Bishop-Phelps-Bollob & aacute;s point and operator properties respectively, where the eta which appears in their definition does not depend just on epsilon > 0 but also on a state (x,x*) or on a numerical radius one operator T. We address the relation between the L-p,L-p-nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and c0 are examples of Banach spaces satisfying the L-p,L-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,L-o-nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.