MATHEMATICA BOHEMICA, Vol. 133, No. 1, pp. 75-83 (2008)

On reflexivity and hyperreflexivity of some spaces of intertwining operators

Michal Zajac

Michal Zajac, Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovicova 3, 812 19 Bratislava 1, Slovak Republic, e-mail:

Abstract: Let $T,T'$ be weak contractions (in the sense of Sz.-Nagy and Foias), $m,m'$ the minimal functions of their $C_0$ 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 $H^2\ominus dH^2$. 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 $I(T,T')$.

Keywords: intertwining operator, reflexivity, $C_0$ contraction, weak contraction, hyperreflexivity

Classification (MSC2000): 47A10, 47A15

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