Abstract
For a bounded linear operator T on a Hilbert space let { T } âČ \{ T\} â and Alg T {\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 Ï ( λ ) \psi (\lambda ) . In this note we prove the equivalence of the following conditions: (i) | Ï ( e i t ) | = 1 |\psi ({e^{it}})| = 1 on a set of positive Lebesgue measure; (ii) Alg T = { T } âČ {\operatorname {Alg}}\;T = \{ T\} â ; (iii) every invariant subspace for T is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy Alg T = { T } âČ {\operatorname {Alg}}\;T = \{T\}â .
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