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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Comparison of commuting one-parameter groups of isometries


Authors: Ola Bratteli, Hideki Kurose and Derek W. Robinson
Journal: Trans. Amer. Math. Soc. 320 (1990), 677-694
MSC: Primary 47D03; Secondary 46L40, 46L57
MathSciNet review: 968886
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0968886-8
Article copyright: © Copyright 1990 American Mathematical Society