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MiS Preprint
41/1997
Conservativeness and extensions of Feller semigroups
Rene Schilling
Abstract
Let $\{T_i\}_i \geq 0$ denote a Feller semigroup on the space of functions vanishing at infinity, $C_{\infty} (R^K)$, and $\{T_i\}_i \geq 0$ 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 $\{T_i\}_i \geq 0$. 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.
