Mean ergodic theorems for bi-continuous semigroups (Q633186)

The notion of bi-continuous semigroups was first introduced and studied by \textit{F. Kühnemund} [Bi-continuous semigroups on spaces with two topologies: theory and applications.\ Ph.\,D.\ Thesis, Universität Tübingen (2001), see also: Semigroup Forum 67, No. 2, 205--225 (2003; Zbl 1064.47040)]. These are exponentially bounded semigroups of bounded linear operators on a Banach space which are locally bi-equicontinuous with respect to a certain locally convex topology weaker than the norm topology, and which are strongly continuous with respect to this topology. In the paper under review, the authors consider the convergence of the Cesàro means of bi-continuous semigroups. They prove several ergodic theorems analogous to the classical ergodic theorems for strongly continuous semigroups. They also apply their results to Feller semigroups generated by autonomous and non-autonomous second-order elliptic differential operators with unbounded coefficients in \(C_{b}(\mathbb R^{N})\).