On the Korovkin Property and Feller Semigroups

Abstract

Let E be a second countable locally compact Hausdorff space and let L be a linear operator with domain D(L) and range R(L) in C ₀ (E) = C_o (E, R). Suppose that the operator L satisfies the maximum principle and that L possesses the Korovkin property in the sense that there exists a strictly positive real number t₀ > 0 such that for every ${{x}_{0}} \in E \cup \{ \Delta \}$ the equality

$$ \begin{gathered} \mathop {\inf }\limits_{h \in D(L)} \mathop {\sup }\limits_{x \in E} \left\{ {h({x_0}) + \left[ {g - \left( {I - {t_0}L} \right)h} \right]\left( x \right)} \right\} \hfill \\ = \mathop {\sup }\limits_{h \in D(L)} \mathop {\inf }\limits_{x \in E} \left\{ {h({x_0}) + \left[ {g - \left( {I - {t_0}L} \right)h} \right]\left( x \right)} \right\} \hfill \\ \end{gathered} $$

is valid for all g∈ C₀ (E). Suppose that the domain D (L) is dense in C₀ (E). Then the operator L extends in a unique fashion to the generator of a Feller semigroup. Moreover, for every x∈ E the martingale problem is well-posed for the operator L.