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MATHEMATICA BOHEMICA, Vol. 133, No. 1, pp. 75-83 (2008)
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On reflexivity and hyperreflexivity of some spaces of intertwining operators

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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: ` michal.zajac@stuba.sk`

**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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