TY - JOUR
T1 - Commutators of composition operators with adjoints of composition operators on weighted Bergman spaces
AU - MacCluer, Barbara D.
AU - Narayan, Sivaram K.
AU - Weir, Rachel J.
N1 - Funding Information:
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.
PY - 2013/1
Y1 - 2013/1
N2 - 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 [Cψ*, Cφ is non-trivially compact on the weighted Bergman space Aα2(D) 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.
AB - 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 [Cψ*, Cφ is non-trivially compact on the weighted Bergman space Aα2(D) 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.
KW - composition operator
KW - essential normality
KW - weighted Bergman space
UR - http://www.scopus.com/inward/record.url?scp=84872051420&partnerID=8YFLogxK
U2 - 10.1080/17476933.2010.551202
DO - 10.1080/17476933.2010.551202
M3 - Article
AN - SCOPUS:84872051420
SN - 1747-6933
VL - 58
SP - 35
EP - 54
JO - Complex Variables and Elliptic Equations
JF - Complex Variables and Elliptic Equations
IS - 1
ER -