Compact Composition Operators on BMOA

We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk $U$ that are Poisson integrals of functions on $\partial U$, 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 $C_\phi: \operatorname{BMOA}\rightarrow\operatorname{BMOA}$ relates to the shape of $\phi(U)$ near $\partial U$, 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.