A geometric approach to finite rank unitary perturbations

TL;DR

This paper develops a geometric framework for analyzing finite rank unitary perturbations of non-unitary contractions, establishing spectral conditions and explicit formulas for characteristic functions, extending classical rank-one results to higher ranks.

Contribution

It introduces a dilation-based method to study finite rank perturbations and derives explicit formulas for characteristic functions in this context, generalizing known rank-one results.

Findings

Spectra are purely singular iff the characteristic function is inner.

Derived explicit formulas for characteristic functions of perturbed contractions.

Extended classical rank-one perturbation results to arbitrary finite rank cases.

Abstract

For a fixed natural number n, we consider a family of rank n unitary perturbations of a completely non-unitary contraction (cnu) with deficiency indices (n,n) on a separable Hilbert space. We relate the unitary dilation of such a contraction to its rank n unitary perturbations. Based on this construction, we prove that the spectra of the perturbed operators are purely singular if and only if the operator-valued characteristic function corresponding to the unperturbed operator is inner. In the case where n=1 the latter statement reduces to a well-known result in the theory of rank one perturbations. However, our method of proof via the theory of dilations extends to the case of arbitrary n. We find a formula for the operator-valued characteristic functions corresponding to a family of related cnu contractions. In the case where n=1, the characteristic function of the original contraction we obtain a simple expression involving the normalized Cauchy transform of a certain measure. An application of this representation then enables us to control the jump behavior of this normalized Cauchy transform "across" the unit circle.