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:X^*→ X be a linear, bounded, non-negative, and symmetric operator and let A:D(A)⊆ X→ X 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, as Q_∞=∫_0+∞ esAQesA^*\,ds, and H_∞ is the Cameron–Martin space associated to μ_∞ . Let H be the reproducing kernel Hilbert space associated with Q with inner product [·,·]_H . We assume that the operator Q_∞ A^*:D(A^*)⊆ X^*→ X extends to a bounded linear operator B(H) which satisfies B+B^*=-Id_H , where Id_H denotes the identity operator on H . Let D and D^2 be the first and second order Fréchet derivative operators. We denote by D_H and (D_H,D^2_H) the closure in L^2(X,μ_∞) of the operators QD and (QD,QD^2) , respectively, defined on smooth cylindrical functions, and by W1,2_H(X,μ_∞) and W2,2_H(X,μ_∞) , respectively, their domains in L^2(X,μ_∞) . Furthermore, we denote by DA_∞ the closure of the operator Q_∞ A^*D in L^2(X,μ_∞) defined on smooth cylindrical functions, and by W1,2A_∞(X,μ_∞) the domain of DA_∞ in L^2(X,μ_∞) . We characterize the domain of the operator L , associated to the bilinear form (u,v)↦-∫X[BD_Hu,D_Hv]_H\,dμ_∞, u,v∈ W1,2_H(X,μ_∞), in L^2(X,μ_∞) . More precisely, we prove that D(L) coincides, up to an equivalent renorming, with a subspace of W2,2_H(X,μ_∞)∩ W1,2A_∞(X,μ_∞) . We stress that we are able to treat the case when L is degenerate and non-symmetric.