A Bishop-Phelps-Bollob\'{a}s theorem for bounded analytic functions

Let $H^\infty$ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by $\mathscr{B}(H^\infty)$ the Banach space of all bounded linear operators from $H^\infty$ to itself. We prove that the Bishop-Phelps-Bollobas property holds for $\mathscr{B}(H^\infty)$. As an application to our approach, we prove that the Bishop-Phelps-Bollobas property also holds for operator ideals of $\mathscr{B}(H^\infty)$.