On the Compact Operators Case of the Bishop–Phelps–Bollobás Property for Numerical Radius

Abstract

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $$C_0(L)$$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $$C_0(L)$$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $$\ell _1$$ have the BPBp-nu for compact operators.