On the Malliavin Calculus for SDE's on Hilbert Spaces

We consider a class of SDE's on Hilbert spaces and study the partial hypoellipticity of generators associated with these SDE's. We show that the Malliavin calculus with a new key lemma is efficient for the purpose. The partial Hörmander theorem is proved in this paper, and it is applied to the problem of propagation of absolute continuity of measures by stochastic flows given by those SDE's.