Comparison of commuting one-parameter groups of isometries

Abstract:

Let $\alpha ,\;\beta$ be two commuting strongly continuous one-parameter groups of isometries on a Banach space $\mathcal {A}$ with generators ${\delta _\alpha }$ and ${\delta _\beta }$, and analytic elements $\mathcal {A}_\omega ^\alpha ,\;\mathcal {A}_\omega ^\beta$, respectively. Then it is easy to show that if ${\delta _\alpha }$ is relatively bounded by ${\delta _\beta }$, then $\mathcal {A}_\omega ^\beta \subseteq \mathcal {A}_\omega ^\alpha$, 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 ${C^{\ast }}$-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.