Commutators of composition operators with adjoints of composition operators on weighted Bergman spaces

Abstract

For linear-fractional self-maps ϕ and ψ of the unit disc 𝔻, where at least one of ϕ and ψ is a non-automorphism, we show that the commutator is non-trivially compact on the weighted Bergman space if and only if either ϕ and ψ are both parabolic or ϕ and ψ are both hyperbolic, with associated conclusions about their fixed points in each case. In the automorphism case, we show that this commutator is compact if and only if both ϕ and ψ are rotations.

Keywords:

• composition operator
• weighted Bergman space
• essential normality

AMS Subject Classifications::

• 47B33
• 30H20

Acknowledgements

The second author thanks Central Michigan University for the support during his Fall 2009 sabbatical leave and University of Virginia for the hospitality extended during his sabbatical visit. The third author thanks the Allegheny College Academic Support Committee for funding provided during the development of this article. The authors also thank Katie Quertermous for helpful discussion related to Corollary 3.8.