This is an announcement for the paper "An example of a minimal action of the free semi-group $\F^{+}_{2}$ on the Hilbert space" by Sophie Grivaux and Maria Roginskaya.

Abstract: 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 ${T^{n}x;\ n\ge 0}$ of every non-zero vector $x\in H$ 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 $x\in H$, either $x$ or $Vx$ 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 $H$.

Archive classification: math.FA math.DS

Remarks: 10 p

Submitted from: grivaux@math.univ-lille1.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1301.6144

or

http://arXiv.org/abs/1301.6144