On the Bishop-Phelps-Bollobas theorem for operators and numerical radius

Abstract

We study the Bishop-Phelps-Bollobas property for numerical radius (for short, BPBp-nu) of operators on l(1)-sums and l(infinity)-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X circle plus(1) Y has the weak BPBp-nu, then (X, Y) has the Bishop-Phelps-Bollobas property (BPBp). On the other hand, if Y is strongly lush and X circle plus(infinity) Y has the weak BPBp-nu, then (X, Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L-1(mu) spaces, and finite-codimensional subspaces of C[0, 1].