Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + <Ax, Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then $$\n P(t)-P(s)\n_{L^1(E,\mu)} = 2$$ for all $t,s\ge 0$ with $t\not=s$. This result is new even when $E = \R^n$. We also study the behaviour of $P$ in the space $BUC(E)$. We show that if $A\not=0$ there exists $t_0>0$ such that $$\n P(t)-P(s)\n_{BUC(E)} = 2$$ for all $0\le t,s\le t_0$ with $t\not=s$. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either $$ \n P(t)- P(s)\n_{BUC(E)} = 2$$ for all $t,s\ge 0$, \ $t\not=s$, or $S$ is the direct sum of a nilpotent semigroup and a finite-dimensional periodic semigroup. Finally we investigate the spectrum of $L$ in the spaces $L^1(E,\mu)$ and $BUC(E)$.