Weakly compact composition operators on vector-valued BMOA

Abstract

Weak compactness of the analytic composition operator f ↦ f ○ φ is studied on BMOA ( X ) , the space of X-valued analytic functions of bounded mean oscillation, and its subspace VMOA ( X ) , where X is a complex Banach space. It is shown that the composition operator is weakly compact on BMOA ( X ) if X is reflexive and the corresponding composition operator is compact on the scalar-valued BMOA. A concrete example is given which shows that BMOA ( X ) differs from the weak vector-valued BMOA for infinite dimensional Banach spaces X.