Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(4): 799-817

Online first article May 31, 2021      Printed July 1, 2021

https://doi.org/10.4134/JKMS.j190722

Copyright © The Korean Mathematical Society.

Normal complex symmetric weighted composition operators on the Hardy space

Hang Zhou, Ze-Hua Zhou

Tianjin Chengjian University; Tianjin University

Abstract

In this paper, we investigate the normal and complex symmetric weighted composition operators $W_{\psi,\varphi}$ on the Hardy space $H^2(\mathbb{D})$. Firstly, we give the explicit conditions of weighted composition operators to be normal and complex symmetric with respect to conjugations $\mathcal{C}_1$ and $\mathcal{C}_2$ on $H^2(\mathbb{D})$, respectively. Moreover, we particularly investigate the weighted composition operators $W_{\psi,\varphi}$ on $H^2(\mathbb{D})$ which are normal and complex symmetric with respect to conjugations $\mathcal{J}$, $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, when $\varphi$ has an interior fixed point, $\varphi$ is of hyperbolic type or parabolic type.

Keywords: Normality, complex symmetric, weighted composition operators, automorphism, Hardy space

MSC numbers: Primary 47B33; Secondary 47B32, 47A05

Supported by: The work was supported in part by the National Natural Science Foundation of China (Grant Nos.11771323;11801402).