Small composition operators on analytic vector-valued function spaces

Let phi be an analytic mapping of the unit disk D into itself. We characterize the weak compactness of the composition operator C-phi : f bar right arrow f circle phi 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 phi in each case. Assuming this, the operator C phi : H-1(X) --> H-1(X) is weakly compact if and only if phi satisfies the Shapiro condition: N-phi(omega) = o(1 - /omega/) as /omega/ --> 1(-), where N-phi stands for the Nevanlinna counting function of phi. This extends a previous result of Sarason in the scalar case. Similarly, C-phi is weakly compact on B1(X) if and only if the angular derivative condition lim(/omega/-->1) - (1- /phi(omega)/)/(l - /omega/) = CO is satisfied. We also characterize the weak compactness of C-phi on vector-valued (little and big) Bloch spaces and on HCO(X). Finally, we find conditions for weak conditional compactness of C-phi on the above mentioned spaces of analytic vector-valued functions.