On the Bishop–Phelps–Bollobás theorem for operators and numerical radius

We study the Bishop–Phelps–Bollobas property for numerical radius (for short, BPBp-nu) of operators on `1-sums and `∞-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 ⊕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 ⊕∞ Y has the weak BPBp-nu, then (X,Y ) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L1(μ) spaces, and finite-codimensional subspaces of C[0, 1].