  A particular case of results from [K2] is as follows. Let the unitary
asymptote of a contraction $T$ contain the bilateral shift (of finite or
infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of
$T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same
multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for
contractions. In the present paper this result is generalized to polynomially
bounded operators, but in the simplest case. Namely, it is proved that if the
unitary asymptote of a polynomially bounded operator $T$ contains the bilateral
shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$
of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of
multiplicity $1$. The proof is based on a result from [B].
