A very particular case of results by Kerchy (2007) is as follows. Let the unitary
asymptote of a contraction T contain the bilateral shift of multiplicity 1. Then there
exists an invariant subspace M of T such that T vertical bar(M) is similar to the
unilateral shift of multiplicity 1. 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. The proof is based on a result by Bourgain (1984).