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.