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

Journal article, 2013

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.