Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||.
A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that Warning: This terminology is not uniform across the literature.
Second, it is more suitable to the treatment of spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.
A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the Hille–Yosida theorem.
The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by It can be shown that it satisfies the identity Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and