The author investigates the Ornstein-Uhlenbeck operator \(L^{\Omega}\) defined as the self-adjoint operator associated to the quadratic form \[ \int_{\Omega}\langle \nabla_Hu,\nabla_Hv\rangle_H\,\gamma(dx),\quad u,\,v\in W^{1,2}(\Omega,\gamma), \] where \(\Omega\) is an open convex subset of an infinite-dimensional separable Banach space \(X\) endowed with a centered non-degenerate Gaussian measure, \(W^{1,2}(\Omega,\gamma)\) is the Sobolev space, and \(\nabla_H\) is the gradient along the Cameron-Martin space \(H\). By using the cylindrical approximation of \(\Omega\), he approximates \(L^{\Omega}\) by finite-dimensional Ornstein-Uhlenbeck operators, and proves some inequalities for the semigroup \((T^{\Omega}(t))_{t\ge 0}\) generated by \(L^{\Omega}\). Then the author proves that \((T^{\Omega}(t))_{t\ge 0}\) is a submarkovian semigroup, and shows the Poincaré inequality and the logarithmic-Sobolev inequality. Spectral properties of \(L^{\Omega}\) are finally deduced.