Trotter-Kato theorems for bicontinuous semigroups and applications to Feller semigroups

In the last years, the study of transition Markov semigroups on spaces of bounded continuous (uniformly continuous) functions led to consider a class of semigroups of operators for which the usual strong continuity fails to hold. For instance, the Ornstein-Uhlenbeck semigroup on the space of uniformly continuous functions on $\R^N$ or even the heat semigroup on the space of bounded continuous functions on $\R^N$ are not $C_0$-semigroups with respect to the sup-norm. It was then natural to look for suitable locally convex topologies weaker than the norm topology to treat the lack of strong continuity. The results of this paper are given in the general framework introduced by Kuhnemund in this direction. In particular, she introduced the so-called bi-continuous semigroups, i.e., semigroups of bounded linear operators on a Banach space which are locally bi-equicontinuous with respect to an additional topology $\tau$ coarser than the norm topology and such that the orbit maps $t\to T(t)x$ are continuous with respect to $\tau$. She obtains generation theorems and also approximation theorems of Trotter-Kato type. In this paper we improve the results on the convergence of bi-continuous semigroups given by Kuhnemund, by relaxing some assumptions, thus answering to an open question asked by Kuhnemund. More precisely, we give a complete characterization of the convergence of semigroups with respect to the topology $\tau$, similar to the classical theorems for $C_0$-semigroups. As a consequence, we obtain a Lie-Trotter product formula and apply it to Feller semigroups generated by second order elliptic differential operators with unbounded coefficients in $C_b(\R^N)$.