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Linear-Fractional Composition Operators in Several Variables

Authors:	Barbara D. MacCluer Rachel J. Weir

Affiliation:	(1) Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA;(2) Present address: Department of Mathematics, Allegheny College, Meadville, PA 16335, USA

Abstract:	We investigate properties of linear-fractional composition operators C_φ on Hardy and Bergman spaces of the ball in $mathbb{C}^N$ that are motivated by a formula for the self-commutator [C_φ^* ,C_φ]. In particular, we characterize when certain commutators [C_φ, C_σ] are compact, and give conditions under which $left[ {T_{z^beta } ^* ,C_varphi } right]$ is compact, where $T_{z^beta }$ 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.

Keywords:	47B33
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