Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions

For a fixed nonnegative integer m, an analytic map phi and an analytic function, the generalized integration operator I-phi,Psi((m)) is defined by I-phi,psi((m)) f (z) = integral(z)(0) f((m))(phi(zeta))psi(zeta) d zeta for X -valued analytic function f, where X is a Banach space. Some estimates for the norm of the operator I-phi,psi((m)) : wA(alpha)(p) (X) -> A(alpha)(p) (X) are obtained. In particular, it is shown that the Volterra operator J(b) : wA(alpha)(p) (X) -> A(alpha)(p) (X) is bounded if and only if J(b) : A(alpha)(2) -> A(alpha)(2) is in the Schatten class Sp(A(alpha)(2)) for 2 <= p < infinity and alpha > 1. Some corresponding results are established for X-valued Hardy spaces and X -valued Fock spaces.