Linear-fractional composition operators in several variables

Abstract

We investigate properties of linear-fractional composition operators C φ on Hardy and Bergman spaces of the ball in ℂN that are motivated by a formula for the self-commutator [C φ *, Cφ]. In particular, we characterize when certain commutators [C φ, C σ] are compact, and give conditions under which [Tzβ*,_Cφ] is compact, where Tzβ is multiplication by the monomial z β. Our results allow us to determine when C φ is essentially normal, for φ belonging to a large class of linear-fractional symbols. © 2005 Birkhäuser Verlag Basel/Switzerland.