Title :

An example of a minimal action of the free semi-group F_2^+ on the Hilbert space

Author(s) :

Grivaux, Sophie [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Roginskaya, Maria [Auteur]
Chalmers University of Technology [Göteborg]

Journal title :

Mathematical Research Letters

Pages :

695 - 704

Publisher :

International Press

Publication date :

2013

ISSN :

1073-2780

English keyword(s) :

Invariant Subspace and Invariant Subset Problems on Hilbert spaces
hy- percyclic vectors
orbits of linear operators
Read’s type operators
minimal action of groups on a Hilbert space.

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

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 {T^n x; n ≥ 0} of every non-zero vector ...
Show more >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 {T^n x; 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.Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL