Bilateral weighted shift operators similar to normal operators

TL;DR

This paper characterizes when an injective bilateral weighted shift operator on () is similar to a normal operator, showing it is equivalent to being similar to a scalar multiple of the unweighted bilateral shift.

Contribution

It provides a precise criterion for similarity to normal operators for a class of bilateral weighted shift operators, extending understanding of their structure.

Findings

Injective bilateral weighted shift operators are similar to normal operators iff similar to scalar multiples of the unweighted shift.

The result characterizes the structure of such operators in terms of similarity transformations.

The paper establishes a necessary and sufficient condition for normality similarity in this operator class.

Abstract

We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$.