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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 $T$. 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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 $T$. 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