Clark measures and de Branges-Rovnyak spaces in several variables

Abstract

Let $B_n$ denote the unit ball of ${\mathbb{C}}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b:\mathcal{D}\to B_1$, we study the corresponding family ${\sigma }_{\alpha }[b]$, $\alpha \in \partial B_1$, of Clark measures on the distinguished boundary $\partial \mathcal{D}$. We construct a natural unitary operator from the de Branges-Rovnyak space $\mathcal{H}(b)$ onto the Hardy space $H^2({\sigma }_{\alpha })$. As an application, for $\mathcal{D}=B_n$ and an inner function $I:B_n\to B_1$, we show that the property ${\sigma }_1[I]\ll {\sigma }_1[b]$ is directly related to the membership of an appropriate explicit function in $\mathcal{H}(b)$.

8 pages; title is changed, Section 1 is rewritten, Section 3 is enlarged