Absolutely continuous spectrum preservation: A new proof for unitary operators under finite-rank multiplicative perturbations

Pablo A. Díaz

doi:10.56754/0719-0646.2703.701

Authors

Pablo A. Díaz Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Avenida Libertador Bernardo O’Higgins nº 3363, Santiago, Chile. https://orcid.org/0000-0002-2365-3245

DOI:

https://doi.org/10.56754/0719-0646.2703.701

Keywords:

Absolutely continuous measure, finite rank perturbations, multiplicative perturbation, unitary operators

Abstract

We will provide a new proof of the Birman-Krein theorem for unitary operators multiplicatively perturbed by finite-rank operators, which is nothing more than the Kato-Rosenblum theorem, but instead of self-adjoint operators. In other words, \(U\) is a unitary operator and \(X\) is a unitary operator given by a finite rank perturbation of the identity, i.e., \(X=\mathbf{1}+W\) with \(W\) finite rank. We show that \(U\) and its perturbed version \(UX\) (or \(XU\)) are unitarily equivalent on their absolutely continuous subspaces.

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