Weighted composition operators on Bers-type spaces of Loo-Keng Hua domains
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 583-595
Published online April 22, 2020
Printed May 31, 2020
Zhi-jie Jiang, Zuo-an Li
Sichuan University of Science and Engineering; Sichuan University of Science and Engineering
Abstract : Let $\mbox{HE}_I$, $\mbox{HE}_{II}$, $\mbox{HE}_{III}$ and $\mbox{HE}_{IV}$ be the first, second, third and fourth type Loo-Keng Hua domain respectively, $\vp$ a holomorphic self-map of $\mbox{HE}_I$, $\mbox{HE}_{II}$, $\mbox{HE}_{III}$, or $\mbox{HE}_{IV}$ and $u\in H(\M)$ the space of all holomorphic functions on $\M\in\{\mbox{HE}_I, \mbox{HE}_{II}, \mbox{HE}_{III}, \mbox{HE}_{IV}\}$. In this paper, motivated by the well known Hua's matrix inequality, first some inequalities for the points in the Bers-type spaces of the Loo-Keng Hua domains are obtained, and then the boundedness and compactness of the weighted composition operators $W_{\vp,u}:f\mapsto u\cdot f\circ\vp$ on Bers-type spaces of these domains are characterized.
Keywords : Weighted composition operator, Hua's matrix inequality, Loo-Keng Hua domain, Bers-type space, boundedness, compactness
MSC numbers : Primary 42A18; Secondary 47B33
Supported by : This work was supported by the Sichuan Science and Technology Program (2018JY0200) and the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2016QZJ01).
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