On the domain of non-symmetric and, possibly, degenerate Ornstein-Uhlenbeck operators in separable Banach spaces.

Let X be a separable Banach space and let X  be its topological dual. Let Q be a linear, bounded, non-negative, and symmetric operator and let A be the infinitesimal generator of a strongly continuous semigroup of contractions on X. We consider the abstract Wiener space (X,µ,H), where µ is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally and H is the Cameron–Martin space associated to µ. Let H be the reproducing kernel Hilbert space associated with Q. We assume that the operator Q extends to a bounded linear operator B which satisfies B+B^*=-IdH, where IdH denotes the identity operator on H. Let D and D2 be the first and second order Fréchet derivative operators. We denote by DH and DH;D2 L2(Xµ) of the operators QD and .QD;QD2/, respectively, defined on smooth cylindrical functions, and by W ^{1,2}_H(X,µ) and W^{2,2}_H(X,µ) respectively, their domains in L2(X,µ). Furthermore, we denote by DA1 the closure of the operator Q1A D in L2(X,µ) defined on smooth cylindrical functions, and by W^{1,2}_{A}(X,µ) the domain of DA1 in L2(X,µ). We characterize the domain of the operator L; more precisely, we prove that D.L coincides, up to an equivalent renorming, with a subspace of W^{2,2}_H(X,µ)\cap W^{1,2}_{A}(X,µ). We stress that we are able to treat the case when L is degenerate and non-symmetric.