Composition operators on vector-valued harmonic functions and Cauchy transforms

Let ' be an analytic self-map of the unit disk. The weak compactness of the composition operators C' : f 7! f ' is characterized on the vector-valued harmonic Hardy spaces h1(X), and on the spaces CT(X) of vector-valued Cauchy transforms, for reexiv e 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' on certain spaces wh1(X) and wCT(X) of weak type by extending an alternative approach due to Bonet, Doma«ski and Lindstrm. Concrete examples based on minimal prerequisites highlight the dierences between hp(X) (respectively, CT (X)) and the corresponding weak spaces.