Essential norm of weighted composition operators from \(H^\infty\) to the Zygmund space


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\(\|uC_\varphi\|_{e,H^\infty\rightarrow \mathcal{Z}} \approx \limsup_{n\rightarrow\infty}\|u\varphi^n\|_\mathcal{Z}\).


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