Unitary Perturbations of Compressed n-Dimensional Shifts

摘要

Given any contractive matrix-valued analytic function Θ on the unit disc D, we construct a u(n)-parameter family of unitary operators which correspond to Θ 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 Θ, 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 Θ to contain a total orthogonal set of point evaluation vectors.