Unitary perturbations of compressed N-dimensional shifts

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