Composition operators on vector-valued harmonic functions and Cauchy transforms

Abstract

Let (p be an analytic self-map of the unit disk. The weak compactness of the composition operators Cq,: f → f o φ is characterized on the vector-valued harmonic Hardy spaces h 1 (X), and on the spaces CT(X) of vector-valued Cauchy transforms, for reflexive Banach spaces X. This provides a vector-valued analogue of results for composition operators which are due to Sarason, Shapiro and Sundberg, as well as Cima and Matheson. We also consider the operators C φ on certain spaces wh 1 (X) and w CT(X) of weak type by extending an alternative approach due to Bonet, Domanski and Lindstrom. Concrete examples based on minimal prerequisites highlight the differences between h p (X) (respectively, CT(X)) and the corresponding weak spaces.