Convergence criteria for a sequence of semi-groups

Abstract

Let A _j be a sequence of infinitesimal generators of contraction semi-groups on a Banach space X whose resolvents converge strongly:I-A _J ¹?R. We show that if R is injective, then the semi-groups converge strongly on the closure of the range of Rin X. In the reflexive Banach space case, the range of R is dense. Injectivity of R is equivalent to "graph convergence" of the sequencE Aj. We give simple proofs of known results of Trotter and of Glimm and Jaffe.