Diffusion semigroups on abstract Wiener space

The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form <!-- MATH $\partial u/\partial t = {L_x}u$ --> was previously established by the author. The present paper investigates the relationship between and the infinitesimal generator <!-- MATH $\mathcal{U}$ --> of the semigroup. In particular, it is shown that <!-- MATH $\mathcal{U}$ --> is the closure of in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.