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Hyper-reflexivity of isometries and weak contractions. (English) Zbl 0819.47058

Let \(T\) be a bounded operator on a complex separable infinite-dimensional Hilbert space \(\mathcal H\). By \(\text{Lat }T\) we mean the lattice of all invariant subspaces of \(T\) and by \(\text{Hyplat }T\) we mean the lattice of all hyperinvariant subspaces of \(T\), i.e. all subspaces of \(\{T\}'\), the commutant of \(T\). In general, if \(\mathcal A\) is a collection of bounded operators on \(\mathcal H\), then \(\text{Lat }{\mathcal A}\) denotes the lattice of all subspaces of \(\mathcal H\) that are invariant for every member of \(\mathcal A\). \(\text{Alg }{\mathcal A}\) is the smallest weakly closed subalgebra of \({\mathcal B}({\mathcal H})\) generated by \(\mathcal A\). For a collection \(\mathcal L\) of subspaces of \(\mathcal H\) we denote by \(\text{Alg }{\mathcal L}\) the family of all operators on \(\mathcal H\) that leave members of \(\mathcal L\) invariant. We call \({\mathcal A}\subset {\mathcal B}({\mathcal H})\) reflexive if \(\text{Alg }{\mathcal A}= \text{Alg Lat }{\mathcal A}\). An operator \(T\) is called hyper-reflexive if its commutant \(\{T\}'\) is a reflexive algebra.
D. Sarason [Pac. J. Math. 17, 511-517 (1966; Zbl 0171.337)] has shown that any commuting family of normal operators is reflexive. J. A. Deddens [Proc. Am. Math. Soc. 28, 509-512 (1971; Zbl 0213.143)] has shown that every isometry is reflexive.
The emphasis of the present article is on the hyper-reflexivity. It is shown that any unilateral shift is hyper-reflexive. Necessary and sufficient conditions are given so that an isometry is hyper-reflexive. A similar characterization is given for a completely nonunitary weak contraction.

MSC:

47L30 Abstract operator algebras on Hilbert spaces
47A15 Invariant subspaces of linear operators
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