Comparison of commuting one-parameter groups of isometries

Abstract

Let α , β \alpha ,\;\beta be two commuting strongly continuous one-parameter groups of isometries on a Banach space A \mathcal {A} with generators δ α {\delta _\alpha } and δ β {\delta _\beta } , and analytic elements A ω α , A ω β \mathcal {A}_\omega ^\alpha ,\;\mathcal {A}_\omega ^\beta , respectively. Then it is easy to show that if δ α {\delta _\alpha } is relatively bounded by δ β {\delta _\beta } , then A ω β ⊆ A ω α \mathcal {A}_\omega ^\beta \subseteq \mathcal {A}_\omega ^\alpha , and in this paper we establish the inverse implication for unitary one-parameter groups on Hilbert spaces and for one-parameter groups of ∗ ^{\ast } -automorphisms of abelian C ∗ {C^{\ast }} -algebras. It is not known in general whether the inverse implication holds or not, but it does not hold for one-parameter semigroups of contractions.