Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains

Abstract

Given a bounded symmetric domain $D$ in ${\mathbb{C}}^n$, we consider the Clark measures $\hat{I}{¼}_{\hat{I}}\pm$, $\hat{I}\pm \in \mathbb{T}$, associated with a rational inner function $\ddot{I}†$ from $D$ into the unit disc in $\mathbb{C}$. We show that $\hat{I}{¼}_{\hat{I}}\pm =c|\nabla \ddot{I}†{|}^{-1}\ddot{I}{‡}_{\mathrm{b}D\cap \ddot{I}{†}^{-1}(\hat{I}\pm )}\cdot {\mathcal{H}}^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm{b}D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(\hat{I}{¼}_{\hat{I}}\pm )$ the closure of the space of holomorphic polynomials in $L^2(\hat{I}{¼}_{\hat{I}}\pm )$, we characterize the $\hat{I}\pm$ for which $H^2(\hat{I}{¼}_{\hat{I}}\pm )=L^2(\hat{I}{¼}_{\hat{I}}\pm )$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains.

15 pages, no figures