  Let $X$ be a separable Banach space and let $Q:X^*\rightarrow X$ 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
$(X,\mu_\infty,H_\infty)$ where $\mu_\infty$ 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
$H_\infty$ is the Cameron--Martin space associated to $\mu_\infty$.
  Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with
inner product $[\cdot,\cdot]_H$. We assume that the operator $Q_\infty
A^*:D(A^*)\subseteq X^*\rightarrow X$ extends to a bounded linear operator
$B\in \mathcal L(H)$ which satisfies $B+B^*=-{\rm Id}_H$, where ${\rm Id}_H$
denotes the identity operator on $H$. Let $D$ and $D^2$ be the first and second
order Fr\'echet derivative operators, we denote by $D_H$ and $D^2_H$ the
closure in $L^2(X,\mu_\infty)$ of the operators $QD$ and $QD^2$ and by
$W^{1,2}_H(X,\mu_\infty)$ and and $W^{2,2}_H(X,\mu_\infty)$ their domains in
$L^2(X,\mu_\infty)$, respectively,. Furthermore, we denote by $D_{A_\infty}$
the closure of the operator $Q_\infty A^*D$ and by
$W^{1,2}_{A_\infty}(X,\mu_\infty)$ its domain in $L^2(X,\mu_\infty)$. 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 $L^2(X,\mu_\infty)$. More precisely,
we prove that $D(L)$ coincides, up to an equivalent remorming, with a subspace
of $W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty)$. We stress
that we are able to treat the case when $L$ is degenerate and non-symmetric.

Functional Analysis;Analysis of PDEs