A general theorem on the convergence of operator semigroups

In all of these theorems the notion of convergence used in (1-1) and (1-2) has essentially been strong convergence. It is the purpose of the present paper to prove analogous theorems for a certain class of notions of convergence, which will include, in the case of Banach spaces of functions with the sup norm, bounded, pointwise convergence, and convergence of bounded sequences which is uniform over compact sets. To motivate the eventual abstract formulation of our problem, let us consider the setting, due to Trotter, which was used in [4] and [7]. Here the Ln and L are Banach spaces. We assume there exist continuous linear maps Pn: L -> Ln such that