Mathematical Research Letters

Volume 20 (2013)

Number 4

An example of a minimal action of the free semi-group $\mathbb{F}^{+}_{2}$ on the Hilbert space

Pages: 695 – 704

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n4.a7

Authors

Sophie Grivaux (CNRS, Laboratoire Paul Painlevé, UMR 8524, Université Lille 1, Villeneuve d’Ascq, France)

Maria Roginskaya (Department of Mathematical Sciences, Chalmers University of Technology, Göteborg, Sweden; Department of Mathematical Sciences, Göteborg University, Göteborg, Sweden)

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 $\mathbb{F}^{+}_{2}$ 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.

Keywords

invariant subspace and invariant subset problems on Hilbert spaces, hypercyclic vectors, orbits of linear operators, Read’s type operators, minimal action of groups on a Hilbert space

2010 Mathematics Subject Classification

37B05, 47A15, 47A16

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