Compact Composition Operators on BMOA

Abstract

We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk U U that are Poisson integrals of functions on ∂ U \partial U , that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator C ϕ : BMOA → BMOA C_\phi : \operatorname {BMOA}\rightarrow \operatorname {BMOA} relates to the shape of ϕ ( U ) \phi (U) near ∂ U \partial U , introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.