On Reflexivity and Hyperreflexivity of Some Spaces of Intertwining Operators

Let T, T ′ be weak contractions (in the sense of Sz.-Nagy and Foiaş), m, m the minimal functions of their C0 parts and let d be the greatest common inner divisor of m, m . It is proved that the space I(T, T ) of all operators intertwining T, T ′ is reflexive if and only if the model operator S(d) is reflexive. Here S(d) means the compression of the unilateral shift onto the space H⊖dH. In particular, in finite-dimensional spaces the space I(T, T ) is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of T, T ′ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of I(T, T ).