One dimensional perturbations of unitaries that are quasiaffine transforms of singular unitaries, and multipliers between model spaces

TL;DR

This paper investigates the structure of certain operators related to singular unitaries, showing they can be represented as multipliers between model spaces and establishing conditions under which such operators are similar to unitary operators.

Contribution

It demonstrates that operators intertwining cyclic singular unitaries with their one-dimensional perturbations are multiplication operators by multipliers between model spaces, and characterizes when these operators are similar to unitaries.

Findings

Intertwining operators are multipliers between model spaces.

Power bounded perturbations of singular unitaries are similar to unitaries.

Provides bounds on the norms of inverse powers of such operators.

Abstract

It is shown that, under some natural additional conditions, an operator which intertwines one cyclic singular unitary operator with one dimensional perturbation of another cyclic singular unitary operator is the operator of multiplication by a multiplier between model spaces. Using this result, it is shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator, and $\sup_{n\geq 0}\|T^{-n}\|\leq(2(\sup_{n\geq 0}\|T^n\|)^2+1)\cdot(\sup_{n\geq 0}\|T^n\|)^5$.