  The question if polynomially bounded operator is similar to a contraction was
posed by Halmos and was answered in the negative by Pisier. His counterexample
is an operator of infinite multiplicity, while all its restrictions on
invariant subspaces of finite multiplicity are similar to contractions. In
[G16], cyclic polynomially bounded operators which are not similar to
contractions was constructed. The construction was based on a perturbation of
the sequence of finite dimensional operators which is uniformly polynomially
bounded, but is not uniformly completely polynomially bounded, constructed by
Pisier. In this paper, a cyclic polynomially bounded operator $T_0$ such that
$T_0$ is not similar to a contraction and $\omega_a(T_0)=\mathbb O$, is
constructed. Here $\omega_a(z)=\exp(a\frac{z+1}{z-1})$, $z\in\mathbb D$, $a>0$,
and $\mathbb D$ is the open unit disk. To obtain such $T_0$, a slight
modification of the construction from [G16] is needed.

Functional Analysis