Contraction semi-groups in a function space

Abstract

Using the concepts of a semi inner-product and a dissipative operator, it is proven that if X is a complex Banach space (under the supremum norm) of bounded complex valued functions on a set S, p is a bounded positive function on S which is bounded away from zero, pX c X, and A is the infinitesimal generator of a strongly continuous (class (C o )) semi-group of contraction operators in X, then pA is also the infinitesimal generator of such a semi-group.