The authors study transition semigroups associated with the stochastic linear Cauchy problem \[ dX(t)=AX(t)+dW_{H}(t), \;t\geq 0,\quad X(0)=x. \] It is assumed that \(A\) is the generator of a \(C_0\)-semigroup \({\mathbf S}=\{S(t)\}_{t\geq 0}\) of bounded linear operators on the separable real Banach space \(E\) and \({\mathbf W}_{H}=\{W_{H}(t)\}_{t\geq 0}\) is a cylindrical Wiener process with the Cameron-Martin space \(H\) which is continuously imbedded into \(E\). If \(E\) is a Hilbert space, a unique solution to the considered problem of the form \(X(t,x)=S(t)x+\int_{0}^{t}S(t-s) dW_{h}(s)\) is called the Ornstein-Uhlenbeck process associated with \(\mathbf S\) and \({\mathbf W}_{h}\). It is a Markov process on \(E\) whose transition semigroup is given by \(P(t)\varphi(x)=\int_{E}\varphi(S(t)x+y) d\mu_{t}(y)\), where \(\{\mu_{t},t\geq 0\}\) is a family of centered Gaussian measures on \(E\) associated with \(\mathbf S\) and \(H\). The aim of this paper is to study, in the case when \(E\) is Banach space, the transition semigroup \(\mathbf P\) and its generator under the assumption that the process \(\{X(t,x)\}_{t\geq 0}\) is well defined and admits an invariant measure \(\mu_{\infty}\). The transition semigroup is studied under the assumption that \(\mathbf S\) restricts to a \(C_0\)-semigroup \({\mathbf S}_{H}\) on \(H\). The authors provide necessary and sufficient conditions for the strong Feller property of \(\mathbf P\) and for the existence of a spectral gap. Conditions for analyticity of \(\mathbf P\) in terms of analyticity of the restricted semigroup \({\mathbf S}_{h}\) are also obtained.