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AN EXAMPLE OF A MINIMAL ACTION OF THE FREE SEMI-GROUP F+2 ON THE HILBERT SPACE

Sophie Grivaux ; Maria Roginskaya (Institutionen för matematiska vetenskaper, matematik)
Mathematical Research Letters (1073-2780). Vol. 20 (2013), 4, p. 695-704.
[Artikel, refereegranskad vetenskaplig]

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator T on a separable infinite-dimensional Hilbert space H such that the orbit {Tnx; n ≥ 0} of every non-zero vector x ∈ H under the action of T is dense in H. We show that there exists a bounded linear operator T on a complex separable infinite-dimensional Hilbert space H and a unitary operator V on H, such that the following property holds true: for every non-zero vector x ∈ H, either x or V x has a dense orbit under the action of T. As a consequence, we obtain in particular that there exists a minimal action of the free semi-group with two generators F+ 2 on a complex separable infinite-dimensional Hilbert space H. The proof involves Read’s type operators on the Hilbert space, and we show in particular that these operators — which were potential counterexamples to the Invariant Subspace Problem on the Hilbert space — do have non-trivial invariant closed subspaces.



Denna post skapades 2013-12-30. Senast ändrad 2014-09-29.
CPL Pubid: 190767

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Matematisk analys

Chalmers infrastruktur