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Keywords:
intertwining operator; reflexivity; $C_0$ contraction; weak contraction; hyperreflexivity
intertwining operator; reflexivity; $C_0$ contraction; weak contraction; hyperreflexivity
Summary:
Let $T,T^{\prime }$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^{\prime }$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^{\prime }$. It is proved that the space $I(T,T^{\prime })$ of all operators intertwining $T,T^{\prime }$ 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 $H^2\ominus dH^2$. In particular, in finite-dimensional spaces the space $I(T,T^{\prime })$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of $T,T^{\prime }$ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of $I(T,T^{\prime })$.
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