Completely continuous Hankel operators on $H^ \infty$ and Bourgain algebras

Abstract:

Let ${({H^\infty })_b}$ be the Bourgain algebra of ${H^\infty } \subset {L^\infty }$. We prove ${({H^\infty })_b} = {H^\infty } + C$. In particular if $f \in {L^\infty }$ then the Hankel operator ${H_f}$ is a compact map of ${H^\infty }$ into BMO iff whenever ${f_n} \to 0$ weakly in ${H^\infty }$, then $\operatorname {dist}{(}f{f_n},{H^\infty }) \to 0$.