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 (Equation presented); 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 ∊ L(H) 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, DH2) the closure in L2(X, μ∞) of the operators QD and (QD, QD2), respectively, defined on smooth cylindrical functions, and by WH1,2(X, μ∞) and WH2,2(X, μ∞), respectively, their domains in L2(X, μ∞). Furthermore, we denote by DA1 the closure of the operator Q∞A*D in L2(X, μ∞) defined on smooth cylindrical functions, and by WA∞1,2 (X, μ∞) the domain of DA1 in L2(X, μ∞). We characterize the domain of the operator L, associated to the bilinear form (Equation presented), in L2(X, μ∞). More precisely, we prove that D(L)coincides, up to an equivalent renorming, with a subspace of WH2,2(X, μ∞) ✦ WA∞1,2 (X, μ∞). We stress that we are able to treat the case when L is degenerate and non-symmetric.