SMALL COMPOSITION OPERATORS ON ANALYTIC VECTOR-VALUED FUNCTION-SPACES

Let phi be an analytic mapping of the unit disk D into itself. We char acterize the weak compactness of the composition operator C-phi : f ba r 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 condit ion: N-phi(omega) = o(1 - /omega/) as /omega/ --> 1(-), where N-phi st ands for the Nevanlinna counting function of phi. This extends a previ ous result of Sarason in the scalar case. Similarly, C-phi is weakly c ompact on B1(X) if and only if the angular derivative condition lim(/o mega/-->1) - (1- /phi(omega)/)/(l - /omega/) = CO is satisfied. We als o characterize the weak compactness of C-phi on vector-valued (little and big) Bloch spaces and on HCO(X). Finally, we find conditions for w eak conditional compactness of C-phi on the above mentioned spaces of analytic vector-valued functions.