Compact and Weakly Compact Composition Operators on BMOA

Abstract

Any analytic map φ of the unit disc \({\mathbb{D}}\) into itself induces a composition operator C φ on BMOA, mapping \({f \mapsto f \circ \varphi}\), where BMOA is the Banach space of analytic functions \({f\colon \mathbb{D} \to \mathbb{C}}\) whose boundary values have bounded mean oscillation on the unit circle. We show that C φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C φ on VMOA implies compactness even from the Bloch space into VMOA.