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Transactions of the American Mathematical Society

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



On $ n$-parameter discrete and continuous semigroups of operators

Author: James A. Deddens
Journal: Trans. Amer. Math. Soc. 149 (1970), 379-390
MSC: Primary 47.50
MathSciNet review: 0259659
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Abstract: We prove that n commuting operators on a Hilbert space can be uniquely simultaneously extended to doubly commuting coisometric operators if and only if they satisfy certain positivity conditions, which for the case $ n = 1$ state simply that the original operator is a contraction. Our proof establishes the connection between these positivity conditions and the backward translation semigroup on $ {l^2}({Z^{ + n}},\mathcal{K})$. A semigroup of operators is unitarily equivalent to backward translation (or a part thereof) on $ {l^2}({Z^{ + n}},\mathcal{K})$ if and only if the positivity conditions are satisfied and the individual operators are coisometries (or contractions) whose powers tend strongly to zero. Analogous results are proven in the continuous case $ {R^{ + n}}$.

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Keywords: Contraction, Hilbert space, unitary dilation, coisometric extension, unilateral shift, backward translation
Article copyright: © Copyright 1970 American Mathematical Society