Contractions with a unilateral shift summand are reflexive
A bounded linear operator T on a complex, separable Hilbert space H is reflexive if Alg T\(=Alg Lat T\), where Alg T and Alg Lat T denote, respectively, the weakly closed algebra generated by T and I, and the weakly closed algebra consisting of those operators which leave invariant all the invariant subspaces of T. Generalizing Deddens' result that isometries are reflexive, the author shows that the direct sum of a unilateral shift and any contraction is reflexive. Some further generalizations are also obtained. This paper concludes with a conjecture that if T is a contraction and there exists an operator \(W\neq 0\) such that \(WT=SW\) for some unilateral shift S, then T is reflexive. It has since been verified by \textit{H. Bercovici} and \textit{K. Takahashi} [J. Lond. Math. Soc., II. Ser. 32, 149-156 (1985; Zbl 0536.47009)].
weakly closed algebra consisting of those operators which leave invariant all the invariant subspaces
direct sum of a unilateral shift and any contraction is reflexive
scientific article; zbMATH DE number 3931816
scientific article