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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 with the bicommutant property


Author: Katsutoshi Takahashi
Journal: Proc. Amer. Math. Soc. 93 (1985), 91-95
MSC: Primary 47A45; Secondary 47A65, 47C05
MathSciNet review: 766534
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Abstract: It is shown that if $ T$ is a contraction for which there is an operator $ W$ with dense range such that $ WT = SW$ for some unilateral shift $ S$, then $ T$ has the bicommutant property, that is, the double commutant of $ T$ is the weakly closed algebra generated by $ T$ and the identity. As an example of such a contraction we have a contraction $ T$ such that $ I - {T^ * }T$ is of trace class and the spectrum of $ T$ fills the unit disc.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0766534-7
PII: S 0002-9939(1985)0766534-7
Keywords: Contraction, the bicommutant property, double commutant, unilateral shift, Hilbert-Schmidt defect operator
Article copyright: © Copyright 1985 American Mathematical Society