Unitary perturbations of compressed N-dimensional shifts

TL;DR

This paper extends the concept of unitary perturbations from scalar to matrix-valued functions, analyzing their properties in higher-dimensional operator spaces and establishing a disintegration theorem for associated measures.

Contribution

It introduces a higher-dimensional analogue of Clark's unitary perturbations for matrix-valued functions and proves a disintegration theorem for related Aleksandrov-Clark measures.

Findings

Unitary perturbations of matrix-valued shifts are characterized in higher dimensions.

A disintegration theorem for matrix-valued Aleksandrov-Clark measures is established.

Perturbations are shown to be unitarily equivalent in the matrix-valued setting.

Abstract

Given a purely contractive matrix-valued analytic function $\Theta$ on the unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary perturbations of the operator $Z_\Theta$ of multiplication by $z$ in the Hilbert space $L^2_\Theta$ of $n-$component vector-valued functions on the unit circle $\bm{T}$ which are square integrable with respect to the matrix-valued measure $\Om_\Theta$ 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 $\bm{M}_n$-valued functions we verify that the $\mc{U} (n)$-parameter family of unitary perturbations of $Z_\Theta ^*$ is unitarily equivalent to a $\mc{U} (n)$-parameter family of unitary perturbations of $X_\Theta$, 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 $K^2_\Theta$, 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 $H^\infty (\bm{D})$... 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 $Z_\Theta$...