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Zbl 1024.47010
Zhou, Zehua; Shi, Jihuai
Compact composition operators on the Bloch space in polydiscs. (English)
[J] Sci. China, Ser. A 44, No.3, 286-291 (2001). ISSN 1006-9283; ISSN 1862-2763/e

Using some results in {\it J.-H. Shi} and {\it L. Lou} [Acta Math. Sin., Engl. Ser. 16, 85-98 (2000; Zbl 0967.32007)], the authors prove that for a holomorphic self-map $\phi=(\phi_1, \cdots, \phi_n)$ of the polydisc $U^n$, the composition operator $C_\phi$ is compact on the Bloch space $\beta(U^n)$ if and only if for every $\varepsilon >0$, there exists a $\delta>0$, such that $$ \sum_{k,l=1}^n \Bigl|\frac{\partial \phi_l(z)}{\partial z_k} \Bigr|\frac{1-|z_k|^2}{1-|\phi_l(z)|^2} < \varepsilon, $$ whenever $\text{dist}(\phi(z), \partial U^n) <\delta$. This is an extension of result by {\it K. Madigan} and {\it A. Matheson} [Trans. Am. Math. Soc. 347, 2679-2687 (1995; Zbl 0826.47023))], to $n \geq 1$.
[Jinkee Lee (Pusan)]

MSC 2000:: *47B33 Composition operators
32A18 Bloch functions etc.

Keywords: Bloch space; polydisc; composition operator; Bergman metric

Citations: Zbl 0967.32007; Zbl 0826.47023

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