Small composition operators on analytic vector-valued function spaces

Abstract

Let be an analytic mapping of the unit disk D into itself. We characterize the weak compactness of the composition operator C : f f on the vector-valued Hardy space H 1 (X) (= H 1 (D, X)) and on the Bergman space B 1 (X), where X is a Banach space. Reflexivity of X is a necessary condition for the weak compactness of C in each case. Assuming this, the operator