J. Korean Math. Soc. 2017; 54(2): 599-612
Online first article December 27, 2016 Printed March 1, 2017
https://doi.org/10.4134/JKMS.j160138
Copyright © The Korean Mathematical Society.
Mahsa Fatehi and Mahmood Haji Shaabani
Shiraz Branch, Islamic Azad University, Shiraz University of Technology
If $\psi$ is analytic on the open unit disk $\mathbb{D}$ and $\varphi$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{\psi,\varphi}$ is defined by $C_{\psi,\varphi}f(z)=\psi(z)f (\varphi (z))$, when $f$ is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^{2}(\beta)$, we prove that if $C_{\psi,\varphi}$ is cohyponormal on $H^{2}(\beta)$, then $\psi$ never vanishes on $\mathbb{D}$ and $\varphi$ is univalent, when $\psi \not \equiv 0$ and $\varphi$ is not a constant function. Moreover, for $\psi=K_{a}$, where $|a| < 1$, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{\psi,\varphi}$. After that, for $\varphi $ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{\psi,\varphi}$, when $\psi \not \equiv 0$ and $\psi$ is analytic on $\overline{\mathbb{D}}$. Finally, we find all normal weighted composition operators which are bounded below.
Keywords: weighted Bergman spaces, Hardy space, weighted composition operator, normaloid operator, cohyponormal operator, normal operator
MSC numbers: Primary 47B33; Secondary 47B15
2021; 58(4): 799-817
2020; 57(4): 973-986
1998; 35(2): 269-280
2000; 37(2): 139-175
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd