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Let a , ß be two commuting strongly continuous one-parameter groups of isometries on a Banach space j/ with generators <5Q and <5" , and analytic elements srff , srff , respectively. Then it is easy to show that if ôa is relatively bounded by <5» , then srff Ç sAf , 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 inversedoi:10.2307/2001696 fatcat:g6u6tw7cb5cqznc2iluyojzviq