Takahashi, Katsutoshi The reflexivity of contractions with nonreductive\( ^ *\)-residual parts. (English) Zbl 0618.47011 Mich. Math. J. 34, 153-159 (1987). 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 H. Bercovici and the author [J. Lond. Math. Soc. II. Ser. 32, 149-156 (1985; Zbl 0536.47009)] and L. Kerchy [Bull. Lond. Math. Soc. 19, 161-166 (1987; Zbl 0594.47007)]. Cited in 3 Documents MSC: 47A45 Canonical models for contractions and nonselfadjoint linear operators 47A15 Invariant subspaces of linear operators 47C05 Linear operators in algebras Keywords:reflexivity; \({}^ *\)-residual part; functional model; \(H^{\infty }\)- functional calculus; contraction on a Hilbert space; bilateral shift Citations:Zbl 0536.47009; Zbl 0594.47007 PDFBibTeX XMLCite \textit{K. Takahashi}, Mich. Math. J. 34, 153--159 (1987; Zbl 0618.47011) Full Text: DOI