On contractions satisfying 𝐴𝑙𝑔𝑇={𝑇}’

Abstract

For a bounded linear operator T on a Hilbert space let { T } ′ \{ T\} ’ 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 ψ ( λ ) \psi (\lambda ) . In this note we prove the equivalence of the following conditions: (i) | ψ ( e i t ) | = 1 |\psi ({e^{it}})| = 1 on a set of positive Lebesgue measure; (ii) Alg T = { T } ′ {\operatorname {Alg}}\;T = \{ T\} ’ ; (iii) every invariant subspace for T is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy Alg T = { T } ′ {\operatorname {Alg}}\;T = \{T\}’ .