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

TL;DR

This paper constructs a specific cyclic polynomially bounded operator that is not similar to a contraction, extending previous work by modifying existing constructions to include a particular spectral property.

Contribution

It introduces a new example of a cyclic polynomially bounded operator not similar to a contraction with a specified spectral measure, refining earlier constructions.

Findings

Constructed a cyclic polynomially bounded operator not similar to a contraction.

The operator has a spectral measure given by , with  defined by an exponential function.

The construction involves a slight modification of previous methods.

Abstract

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.