Clark measures and de Branges–Rovnyak spaces in several variables

Abstract

Let (Formula presented.) denote the unit ball of (Formula presented.), (Formula presented.), and let (Formula presented.) denote a finite product of (Formula presented.), (Formula presented.). Given a non-constant holomorphic function (Formula presented.), we study the corresponding family (Formula presented.), (Formula presented.), of Clark measures on the distinguished boundary (Formula presented.). We construct a natural unitary operator from the de Branges–Rovnyak space (Formula presented.) onto the Hardy space (Formula presented.). As an application, for (Formula presented.) and an inner function (Formula presented.), we show that the property (Formula presented.) is directly related to the membership of an appropriate explicit function in (Formula presented.).