Transfer operators on complex hyperbolic spaces

Abstract

Let Bⁿ be the unit ball in the n-dimensional complex space and let Δ be the Bergman Laplacian on it. For λ ∈ ℂ such that |ℜ(i λ)| < n we give explicitly the transfer operator from the space of holomorphic functions Bⁿ onto an eigenspace E_λ^+ (Bⁿ ) of Δ. This answers a question raised by Eymard in [2]. As application, for λ = − iη with 0 < η < n, we get that the classical Hardy space H²(Bⁿ ) is isometrically isomorphic to the space H_λ^₂ (Bⁿ ) = { F ∈ E_ₙ^⁺(Bⁿ ) : sup 0<r<1 ( 1 − r²) [∫_(∂Bⁿ )⎸F(rƟ)|²dƟ ]½< ∞ }: Consequently H_λ^₂ (Bⁿ ) is a Banach space.