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

Let $X$ be a separable Banach space and let $Q:X^{\ast }\to X$ be a linear, bounded, non-negative and symmetric operator and let $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $X$. We consider the abstract Wiener space $(X,{\mu }_{\mathrm{\infty }},H_{\mathrm{\infty }})$ where ${\mu }_{\mathrm{\infty }}$ is a centred non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as \begin{align*} Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^}ds, \end{align} and $H_{\mathrm{\infty }}$ is the Cameron–Martin space associated to ${\mu }_{\mathrm{\infty }}$. Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with inner product $[\cdot ,\cdot {]}_H$. We assume that the operator $Q_\infty A^:D(A^)\subseteq X^\rightarrow X$extendstoaboundedlinearoperator$B\in \mathcal L(H)$whichsatisfies$B+B^=-{\rm Id}_H$,where${\rm Id}H$denotestheidentityoperatoron$H$\mathrm{.}Let$D$and$D^2$ be the first and second order Fr'echet derivative operators, we denote by $D_H$ and $D_H^2$ the closure in $L^2(X,{\mu }_{\mathrm{\infty }})$ of the operators $QD$ and $Q{D}^2$ and by $W_H^{1,2}(X,{\mu }_{\mathrm{\infty }})$ and and $W_H^{2,2}(X,{\mu }_{\mathrm{\infty }})$ their domains in $L^2(X,{\mu }_{\mathrm{\infty }})$, respectively,. Furthermore, we denote by $D_{A_{\mathrm{\infty }}}$ the closure of the operator $Q_{\mathrm{\infty }}{A}^{\ast }D$ and by $W_{A_{\mathrm{\infty }}}^{1,2}(X,{\mu }_{\mathrm{\infty }})$ its domain in $L^2(X,{\mu }_{\mathrm{\infty }})$. We characterize the domain of the operator $L$, associated to the bilinear form \begin{align*} (u,v)\mapsto-\int{X}[BD_Hu,D_Hv]Hd\mu\infty, \qquad u,v\in W^{1,2}H(X,\mu\infty), \end{align*} in $L^2(X,{\mu }_{\mathrm{\infty }})$. More precisely, we prove that $D(L)$ coincides, up to an equivalent remorming, with a subspace of $W_H^{2,2}(X,{\mu }_{\mathrm{\infty }})\cap W_{A_{\mathrm{\infty }}}^{1,2}(X,{\mu }_{\mathrm{\infty }})$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.