Compact Differences of Composition Operators in Several Variables

Abstract

When φ and ψ are linear–fractional self-maps of the unit ball BN in \({{\mathbb C}^N,N\geq 1}\), we show that the difference \({C_{\varphi}-C_{\psi}}\) cannot be non-trivially compact on either the Hardy space H2(BN) or any weighted Bergman space \({A^2_{\alpha}(B_N)}\). Our arguments emphasize geometrical properties of the inducing maps φ and ψ.