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Complex symmetric composition operators on weighted Hardy spaces


Authors: Sivaram K. Narayan, Daniel Sievewright and Maria Tjani
Journal: Proc. Amer. Math. Soc. 148 (2020), 2117-2127
MSC (2010): Primary 47B33, 47B32, 47B99
DOI: https://doi.org/10.1090/proc/14909
Published electronically: February 12, 2020
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Abstract: Let $ \varphi $ be an analytic self-map of the open unit disk $ \mathbb{D}$. We study the complex symmetry of composition operators $ C_\varphi $ on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of $ \mathbb{D}$ that is not an elliptic automorphism, we establish that if $ C_{\varphi }$ is complex symmetric, then either $ \varphi (0)=0$ or $ \varphi $ is linear. In the case of weighted Bergman spaces $ A^{2}_{\alpha }$, we find the non-automorphic linear fractional symbols $ \varphi $ such that $ C_{\varphi }$ is complex symmetric.


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Additional Information

Sivaram K. Narayan
Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859
Email: sivaram.narayan@cmich.edu

Daniel Sievewright
Affiliation: 5235 S. Chandler Rd., St. Johns, Michigan 48879
Email: dssievewright@gmail.com

Maria Tjani
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: mtjani@uark.edu

DOI: https://doi.org/10.1090/proc/14909
Keywords: Complex symmetric operator, conjugation, composition operator, weighted Hardy space, weighted Bergman space, linear fractional maps.
Received by editor(s): July 26, 2019
Received by editor(s) in revised form: September 30, 2019
Published electronically: February 12, 2020
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2020 American Mathematical Society