Compact difference of composition operators with smooth symbols on the unit ball

A long-standing problem raised by Shapiro and Sundberg in 1990, the characterization for compact differences of composition operators acting on the Hardy space over the unit disk, has been recently obtained in terms of certain Carleson measures ([4]). The function theoretic characterization for compact differences still remains open in the Hardy space case even on the unit disc and the situation is much more complicated for the several variables case. In this article, we investigate a function theoretic characterization of the compact difference on the Hardy or the Bergman spaces over the unit ball. The condition ρ(φ(z),ψ(z))→0 as max⁡{|φ(z)|,|ψ(z)|}→1 is shown to be sufficient for the difference, C_φ−C_ψ, of two composition operators to be compact on the Hardy or the Bergman spaces over the unit ball when each single composition operator is bounded. We show that this condition is also a necessary condition if the symbols are of class Lip₁(B).