A Bishop-Phelps-Bollobás theorem for bounded analytic functions

Abstract

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

Lemma 3.1 is incorrect, so the proof of Theorem 3.2 is wrong. The same applies to the results of Section 4. However, the Urysohn-type lemma in Section 2 remains correct and appears to be a result of independent interest