Journal of the
Korean Mathematical Society
JKMS

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

Article

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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.

Normal, cohyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces

Mahsa Fatehi and Mahmood Haji Shaabani

Shiraz Branch, Islamic Azad University, Shiraz University of Technology

Abstract

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