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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On contractions satisfying $ {\rm Alg}\ T=\{T\}'$


Author: Pei Yuan Wu
Journal: Proc. Amer. Math. Soc. 67 (1977), 260-264
MSC: Primary 47A45; Secondary 47A60
DOI: https://doi.org/10.1090/S0002-9939-1977-0461177-2
MathSciNet review: 0461177
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Abstract: For a bounded linear operator T on a Hilbert space let $ \{ T\} '$ and $ {\operatorname{Alg}}\;T$ denote the commutant, the double commutant and the weakly closed algebra generated by T and 1, respectively. Assume that T is a completely nonunitary contraction with a scalar-valued characteristic function $ \psi (\lambda )$. In this note we prove the equivalence of the following conditions: (i) $ \vert\psi ({e^{it}})\vert = 1$ on a set of positive Lebesgue measure; (ii) $ {\operatorname{Alg}}\;T = \{ T\} '$; (iii) every invariant subspace for T is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy $ {\operatorname{Alg}}\;T = \{T\}'$.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0461177-2
Keywords: Completely nonunitary contractions, characteristic functions, invariant subspaces, hyperinvariant subspaces, commutants, algebras of operators
Article copyright: © Copyright 1977 American Mathematical Society