MiS Preprint Repository

Let $\{T_i{\}}_i\ge 0$ denote a Feller semigroup on the space of functions vanishing at infinity, $C_{\mathrm{\infty }}(R^K)$, and $\{T_i{\}}_i\ge 0$ its (canonical) extension to the bounded measurable functions. We show that $T_{i}1\in C_b({\mathbb{R}}^k)$ is necessary and sufficient for the invariance of $C_b({\mathbb{R}}^k)$ under $\{T_i{\}}_i\ge 0$. If the generator of the semigroup is a pseudo-differential operator we can restate this condition in terms of the symbol. As a by-product, we obtain necessary and sufficient conditions for the conservativeness of the semigroup which are again expressed through the symbol.