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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Contractions without cyclic vectors


Authors: Béla Sz.-Nagy and Ciprian Foiaş
Journal: Proc. Amer. Math. Soc. 87 (1983), 671-674
MSC: Primary 47A15; Secondary 47A20
MathSciNet review: 687638
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Abstract: It is proved that if $ T$ is a completely nonunitary contraction on Hilbert space such that $ {T^{*n}}$ does not converge strongly to 0 as $ n \to \infty $, there is an integer $ N > 0$ so that none of the powers $ {T^{*m}}$ with $ m \geqslant N$ has a cyclic vector. Both conditions on $ T$ are essential, and the integer $ N$ is not universal, i.e., it depends on $ T$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687638-1
PII: S 0002-9939(1983)0687638-1
Keywords: Contraction, cyclic vector, unitary and isometric dilation
Article copyright: © Copyright 1983 American Mathematical Society