On power bounded operators that are quasiaffine tranforms of singular unitaries

TL;DR

This paper investigates whether power bounded operators that are quasiaffine transforms of singular unitaries are necessarily similar to them, providing partial positive results but leaving the general case unresolved.

Contribution

The paper offers new insights into the similarity problem for power bounded operators related to singular unitaries, extending known results in specific cases.

Findings

Positive results in particular cases for the similarity question

The general problem remains open and unresolved

Connections to polynomially bounded operators are discussed

Abstract

In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In this paper a positive answer is given in some particular cases, but in general an answer remains unknown.