Examples of cyclic polynomially bounded operators that are not similar to contractions, II

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.