Compact differences of composition operators induced by smooth self-maps of the complex unit ball

Abstract

Let φ and ψ be holomorphic self-maps of the unit ball B N , which are sufficiently smooth on the sphere ∂ B N . In this paper, the compact differences of composition operators induced by φ and ψ on the weighted Bergman spaces or Hardy spaces are studied. The main result concerning φ and ψ with the same boundary data is a nontrivial generalization of the corresponding results in the case of one complex variable. It reveals the deep connection between the induced maps φ and ψ on the sphere.