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 $\mu _\alpha$, $\alpha \in \mathbb {T}$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb {C}$. We show that $\mu _\alpha =c|\nabla \varphi |^{-1}\chi _{\mathrm b D \cap \varphi ^{-1}(\alpha )}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Šilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(\mu _\alpha )$ the closure of the space of holomorphic polynomials in $L^2(\mu _\alpha )$, we characterize the $\alpha$ for which $H^2(\mu _\alpha )=L^2(\mu _\alpha )$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains.