Trotter--Kato theorems for bi-continuous semigroups and applications to Feller semigroups. (Q1425128)

The authors prove some results of Trotter--Kato type and a Lie--Trotter product formula for a certain class of semigroups, called bi-continuous. Essentially, these are semigroups of linear operators in a Banach space \(X\), where the usual conditions of strong continuity with respect to the basic topology is replaced with strong continuity with respect to a certain locally convex topology which is weaker than the basic one. For example, the basic topology can be the topology of uniform convergence, while the weaker can be the topology of uniform convergence in compact subsets. The abstract results are applied to certain semigroups generated by elliptic operators with possibly unbounded coefficients in \(\mathbb R^n\).