On contractions satisfying $\textrm {Alg}\ T=\{T\}’$

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 ${\operatorname {Alg}}\;T = \{T\}’$.