Strongly continuous semigroups on some Fréchet spaces

Abstract

We prove that for a strongly continuous semigroup T on the Frechet space omega of all scalar sequences, its generator is a continuous linear operator A is an element of L(omega) and that, for all x is an element of omega and t >= 0, the series exp(tA)(x) = Sigma(infinity)(k=0) t(k)/k! A(k)(x) converges to T-t(x). This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces. (C) 2013 Elsevier Inc. All rights reserved.