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Difference of composition operators over the half-plane


Authors: Boo Rim Choe, Hyungwoon Koo and Wayne Smith
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 47B33; Secondary 30H20
Published electronically: September 1, 2016
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Abstract: We study the differences of composition operators acting on
weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is Hilbert-Schmidt.


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Additional Information

Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
Email: cbr@korea.ac.kr

Hyungwoon Koo
Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
Email: koohw@korea.ac.kr

Wayne Smith
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: wayne@math.hawaii.edu

DOI: https://doi.org/10.1090/tran/6742
Keywords: Difference of composition operators, joint pullback measure, Carleson measure, bounded operator, compact operator, Hilbert-Schmidt operator, weighted Bergman space, half-plane
Received by editor(s): April 14, 2014
Received by editor(s) in revised form: April 27, 2015
Published electronically: September 1, 2016
Additional Notes: The first author was supported by NRF(2015R1D1A1A01057685) of Korea
The second author was supported by NRF(2014R1A1A2054145) of Korea
Article copyright: © Copyright 2016 American Mathematical Society