Unitary perturbations of compressed N-dimensional shifts

Abstract

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

Submitted to Compl. Anal. Oper. Theory