On contractions satisfying $\textrm {Alg}\ T=\{T\}’$
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- by Pei Yuan Wu PDF
- Proc. Amer. Math. Soc. 67 (1977), 260-264 Request permission
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) $|\psi ({e^{it}})| = 1$ 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 ${\operatorname {Alg}}\;T = \{T\}’$.References
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Additional Information
- © Copyright 1977 American Mathematical Society
- 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