  We investigate the transition semigroup of the solution to a stochastic
evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is the
generator of a $C_0$-semigroup $S$ on a separable real Banach space $E$ and
$W_H$ is cylindrical white noise with values in a real Hilbert space $H$ which
is continuously embedded in $E$. Various properties of these semigroups, such
as the strong Feller property, the spectral gap property, and analyticity, are
characterized in terms of the behaviour of $S$ in $H$. In particular we
investigate the interplay between analyticity of the transition semigroup,
$S$-invariance of $H$, and analyticity of the restricted semigroup $S_H$.
