Differentiable perturbations of Ornstein-Uhlenbeck operators

We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator $(L,D(L))$ in the space of all uniformly continuous and bounded functions $f:H\to \text{Rset}$, where $H$ is a separable Hilbert space. We consider a perturbation of the form $N_0\varphi =L\varphi +<D\varphi ,F>$ where $F:H\to H$ is bounded and Fr'echet differentiable with uniformly continuous and bounded differential. Hence, we prove that $N_0$ is $m$-dissipative and its closure in $C_b(H)$ coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in $H$.