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

TL;DR

The paper constructs cyclic polynomially bounded operators that are not similar to contractions, expanding understanding of operator similarity and providing new examples related to a longstanding question in operator theory.

Contribution

It introduces new cyclic polynomially bounded operators not similar to contractions, based on perturbations of Pisier's sequence, and explores their quasisimilarity to $C_0$-contractions or isometries.

Findings

Constructed operators are not similar to contractions.

Operators are quasisimilar to $C_0$-contractions or isometries.

Based on perturbations of Pisier's sequence.

Abstract

A question if a 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 the paper, cyclic polynomially bounded operators which are not similar to contractions and are quasisimilar to $C_0$-contractions or to isometries are constructed. The construction is 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.