A geometric approach to finite rank unitary perturbations R. DouglasConstanze Liaw 44A1547A2047A55Finite rank perturbationsrank one perturbationsdilation theorynormalized Cauchy transform 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5028 10.1512/iumj.2013.62.5028 en Indiana Univ. Math. J. 62 (2013) 333 - 354 state-of-the-art mathematics http://iumj.org/access/