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

Volume 9, Issue 7, pp 5082--5092 http://dx.doi.org/10.22436/jnsa.009.07.11

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

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Keywords

• Zygmund space
• essential norm
• weighted composition operator.

MSC

•  47B38
•  30H30

References

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