The reflexivity of contractions with nonreductive\( ^ *\)-residual parts (Q1822321)

A bounded linear operator T on a Hilbert space is said to be reflexive if the algebra of all operators which leave invariant every invariant subspace of T is equal to the weakly closed algebra generated by T and the identity. It is proved that if T is a contraction on a Hilbert space and there exists an operator Y with dense range such that \(YT=WY\) for some bilateral shift W(\(\neq 0)\), then T is reflexive. This result extends results of \textit{H. Bercovici} and the author [J. Lond. Math. Soc. II. Ser. 32, 149-156 (1985; Zbl 0536.47009)] and \textit{L. Kerchy} [Bull. Lond. Math. Soc. 19, 161-166 (1987; Zbl 0594.47007)].