# Essential norm of weighted composition operators from $H^\infty$ to the Zygmund space

Volume 9, Issue 7, pp 5082--5092 Publication Date: July 30, 2016
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### Authors

Qinghua Hu - Department of Mathematics, Shantou University, Shantou, Guangdong, China. Xiangling Zhu - Department of Mathematics, Jiaying University, 514015, Meizhou, Guangdong, China.

### Abstract

Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $u \in H(\mathbb{D})$, the space of analytic functions on $\mathbb{D}$. The weighted composition operator, denoted by $uC_\varphi$, is defined by $(uC_\varphi f)(z) = u(z)f(\varphi(z)); f \in H(\mathbb{D}); z \in \mathbb{D}.$ In this paper, we give three different estimates for the essential norm of the operator $uC_\varphi$ from $H^\infty$ into the Zygmund space, denoted by $\mathcal{Z}$. In particular, we show that$\|uC_\varphi\|_{e,H^\infty\rightarrow \mathcal{Z}} \approx \limsup_{n\rightarrow\infty}\|u\varphi^n\|_\mathcal{Z}$.

### Keywords

• Zygmund space
• essential norm
• weighted composition operator.

•  47B38
•  30H30

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