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

TL;DR

This paper establishes the Bishop-Phelps-Bollobás property for the space of bounded linear operators on the algebra of bounded analytic functions, extending to certain operator ideals, thus advancing the understanding of operator approximation properties.

Contribution

It proves the Bishop-Phelps-Bollobás property for  and its operator ideals, a novel extension in the context of bounded analytic functions.

Findings

Bishop-Phelps-Bollobás property holds for 

Property extends to operator ideals of 

Advances operator approximation theory in complex analysis

Abstract

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-Bollob\'{a}s property holds for $\mathscr{B}(H^\infty)$. As an application to our approach, we prove that the Bishop-Phelps-Bollob\'{a}s property also holds for operator ideals of $\mathscr{B}(H^\infty)$.