On various types of density of numerical radius attaining operators

In this paper, we are interested in studying Bishop–Phelps–Bollobás type properties related to the denseness of operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and second index strictly positive satisfies point property, see one-dimensional is only one both 1 property. also consider two weaker Lp,p-nu Lo,o-nu, local versions operator respectively, where η appears definition does not depend just on ε>0 but state (x,x∗) or radius T. address relation between strong subdifferentiability X. show finite dimensional spaces c0 examples satisfying Lp,p-nu, exhibit an example strongly subdifferentiable failing it. finish paper by showing satisfy Lo,o-nu that, if X has approximation property equivalent dimensionality.