Quadratic integrals and factorization of linear operators

We consider operators on the matrix-valued disc algebra. We show that any bounded linear operator T:A (X) over bar K --> X to a Banach space X of cotype 2 induces a bounded operator G(nu) --> X defined on some weighted Bergman space G(nu) on the unit disc. We give sufficient conditions on the weight nu for the formal inclusion A (X) over bar K --> G(nu) to be 2-C*-summing. Decompositions of T with respect to operator-valued measures are obtained.