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A Bishop-Phelps-Bollobás theorem for bounded analytic functions

Neeru Bala, Kousik Dhara, Jaydeb Sarkar, Aryaman Sensarma

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

A Bishop-Phelps-Bollobás theorem for bounded analytic functions

Abstract

Let denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by the Banach space of all bounded linear operators from to itself. We prove that the Bishop-Phelps-Bollobás property holds for . As an application to our approach, we prove that the Bishop-Phelps-Bollobás property also holds for operator ideals of .

Paper Structure

This paper contains 4 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. A Urysohn-type lemma for $H^\infty$
  3. Proof of the main theorem
  4. Concluding remarks

Key Result

Theorem \oldthetheorem

The set of norm attaining functionals on a Banach space $X$ is norm dense in its dual space $X^*$.

Theorems & Definitions (14)

  • Theorem \oldthetheorem: Bishop and Phelps
  • Theorem \oldthetheorem: Bishop-Phelps-Bollobás
  • Definition \oldthetheorem: Acosta, Aron, García, and Maestre
  • Theorem \oldthetheorem: Main result
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • ...and 4 more