IUMJ

Title: A geometric approach to finite rank unitary perturbations

Authors: Ronald G. Douglas and Constanze Liaw

Issue: Volume 62 (2013), Issue 1, 333-354

Abstract:

This paper concerns certain families of unitary operators, defined on a separable Hilbert space. Each family consists of all rank $n$ perturbations of a given completely nonunitary (cnu) contraction $T$ with defect indices $(n,n)$. We use the highly developed model theory of Sz.-Nagy and Foia\c s. Namely, for fixed $n\in\mathbb{N}$, we consider a family of rank $n$ perturbations of $T$. Moreover, we also consider the analogous family of cnu contractions that arise as rank $n$ perturbations of $T$. We allow the corresponding characteristic operator function of $T$ to be non-inner. 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 $n\in\mathbb{N}$. We find a formula for the operator-valued characteristic functions corresponding to a family of related cnu contractions. In the case where $n=1$, for 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.