Compact Differences of Composition Operators on Bloch and Lipschitz Spaces

Abstract

We consider the difference T = Cφ − Cψ of two analytic composition operators in the unit disc. We characterize the compactness and weak compactness of T on the standard Bloch space, improving an earlier result by Hosokawa and Ohno. We also characterize the compactness and weak compactness of T on analytic Lipschitz spaces. These characterizations are derived from a general result dealing with differences of weighted composition operators on weighted Banach spaces of analytic functions. We also make complementary remarks on the compactness properties of a single composition operator on the Lipschitz spaces and answer a question of Cowen and MacCluer on the boundedness of such an operator.