Bishop-Phelps-Bollobás property for certain spaces of operators

We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ) into Y have the Bishop-Phelps-Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop-Phelps-Bollobás property.