Clark measures and de Branges–Rovnyak spaces in several variables. Issue 2 (1st February 2023) - Aleksandrov, Aleksei B.; Doubtsov, Evgueni
Abstract : Let B n denote the unit ball of C n, n ≥ 1, and let D denote a finite product of B n j, j ≥ 1 . Given a non-constant holomorphic function b : D → B 1, we study the corresponding family σ α [ b ], α ∈ ∂ B 1, of Clark measures on the distinguished boundary ∂ D . We construct a natural unitary operator from the de Branges–Rovnyak space H ( b ) onto the Hardy space H 2 ( σ α ) . As an application, for D = B n and an inner function I : B n → B 1, we show that the property σ 1 [ I ] ≪ σ 1 [ b ] is directly related to the membership of an appropriate explicit function in H ( b ) .