On contractions satisfying

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 and 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 . In this note we prove the equivalence of the following conditions: (i) on a set of positive Lebesgue measure; (ii) ; (iii) every invariant subspace for *T* is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy .

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Additional Information

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