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

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 {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.