Unitary perturbations of compressed N-dimensional shifts

Abstract

Given a purely contractive matrix-valued analytic function $Θ$ on the unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary perturbations of the operator $Z_Θ$ of multiplication by $z$ in the Hilbert space $L^2_Θ$ of $n-$component vector-valued functions on the unit circle $\bm{T}$ which are square integrable with respect to the matrix-valued measure $\Om_Θ$ determined uniquely by $Θ$ and the matrix-valued Herglotz representation theorem. In the case where $Θ$ is an extreme point of the unit ball of bounded $\bm{M}_n$-valued functions we verify that the $\mc{U} (n)$-parameter family of unitary perturbations of $Z_Θ^*$ is unitarily equivalent to a $\mc{U} (n)$-parameter family of unitary perturbations of $X_Θ$, the restriction of the backwards shift in $H^2_n (\bm{D})$, the Hardy space of $\bm{C} ^n$ valued functions on the unit disc, to $K^2_Θ$, the de Branges-Rovnyak space constructed using $Θ$. These perturbations are higher dimensional analogues of the unitary perturbations introduced by D.N. Clark in the case where $Θ$ is a scalar-valued ($n=1$) inner function, and studied by E. Fricain in the case where $Θ$ is scalar-valued and an extreme point of the unit ball of $H^\infty (\bm{D})$... A matrix-valued disintegration theorem for the Aleksandrov-Clark measures associated with matrix-valued contractive analytic functions $Θ$ is obtained as a consequence of the Weyl integration formula for $\mc{U}(n)$ applied to the family of unitary perturbations of $Z_Θ$...