  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 $H$.

Functional Analysis;Dynamical Systems