The Bishop–Phelps–Bollobás theorem for operators from L1(μ) to Banach spaces with the Radon–Nikodým property

Let Y be a Banach space and (Ω,Σ,μ) be a σ-finite measure space, where Σ is an infinite σ-algebra of measurable subsets of Ω. We show that if the couple (L1(μ),Y) has the Bishop–Phelps–Bollobás property for operators, then Y has the AHSP. Further, for a Banach space Y with the Radon–Nikodým property, we prove that the couple (L1(μ),Y) has the Bishop–Phelps–Bollobás property for operators if and only if Y has the AHSP.