Similarity of perturbations of the shift and a different product of rational functions

TL;DR

This paper establishes necessary and sufficient conditions for the similarity of certain rank-one perturbations of the shift operator, using explicit intertwiners parametrized by algebraic elements.

Contribution

It provides a complete characterization of when two such perturbations are similar, with explicit construction of intertwiners based on algebraic parameters.

Findings

Conditions for similarity are characterized explicitly.

Intertwiners are constructed using elements of a specific algebra.

Invertibility of intertwiners relates to 'circle invertible' elements.

Abstract

Necessary and sufficient conditions are given for the similarity between two perturbations of the (backward) shift by rank one operators, under certain assumptions on the perturbations. The proof of similarity is based on an explicit construction of intertwiners between the perturbations. These intertwiners, in turn, are parametrized by the elements of a certain algebra, with the group of "circle invertible" elements of this algebra giving rise to invertible intertwiners.