On existence of shift-type invariant subspaces for polynomially bounded operator

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].