Comparison of commuting one-parameter groups of isometries

Abstract

Let α, β be two commuting strongly continuous one-parameter groups of isometries on a Banach space A with generators δα nd δβ and analytic elements A, A respectively. Then it is easy to show that if δα is relatively bounded by δβ then, and in this paper we establish the inverse implication for unitary one-parameter groups on Hubert spaces and for one-parameter groups of -automorphisms of abelian C*-algebras. It is not known in general whether the inverse implication holds for one-parameter semigroups of contractions.