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

The question if every 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 [G], cyclic polynomially bounded operators which are not similar to contractions were constructed. The construction was based on a perturbation of a sequence of finite-dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, studied earlier by Pisier. In this paper, a cyclic polynomially bounded operator T0 is constructed so that T0 is not similar to a contraction and ωa(T0) = O. Here ωa(z) = exp(a z+1 z−1 ), z ∈ D, a > 0, and D is the open unit disk. To obtain such a T0, a slight modification of the construction from [G] is needed