Unitary Perturbations of Compressed n-Dimensional Shifts

Abstract

Given any contractive matrix-valued analytic function ${\Theta}$ on the unit disc ${\mathbb{D}}$ , we construct a ${\mathcal{U}(n)}$ -parameter family of unitary operators which correspond to ${\Theta}$ in a natural way. These operators are unitarily equivalent to higher dimensional analogues of Clark’s unitary perturbations, a family of unitary operators associated to any self-map of the unit disc. Clark’s unitary perturbations were introduced in a seminal paper of Clark which has inspired the study of what are now called Aleksandrov–Clark measures. Our higher dimensional analogues of Clark’s unitary perturbations are applied to obtain matrix-generalizations of several classical results on the Aleksandrov–Clark measures associated to any holomorphic self-map of the unit disc. In particular we establish a matrix-valued Aleksandrov disintegration theorem for the Aleksandrov–Clark measures associated with matrix-valued contractive analytic functions ${\Theta}$ , and, by following results of Clark and Fricain in the scalar case, we provide a necessary and sufficient condition for the de Branges–Rovnyak space associated with ${\Theta}$ to contain a total orthogonal set of point evaluation vectors.