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Compact composition operators on the Bloch space in polydiscs. (English) Zbl 1024.47010

Using some results in J.-H. Shi and L. Lou [Acta Math. Sin., Engl. Ser. 16, 85-98 (2000; Zbl 0967.32007)], the authors prove that for a holomorphic self-map $\varphi =\left({\varphi }_{1},\cdots ,{\varphi }_{n}\right)$ of the polydisc ${U}^{n}$, the composition operator ${C}_{\varphi }$ is compact on the Bloch space $\beta \left({U}^{n}\right)$ if and only if for every $\epsilon >0$, there exists a $\delta >0$, such that

$\sum _{k,l=1}^{n}\left|\frac{\partial {\varphi }_{l}\left(z\right)}{\partial {z}_{k}}\right|\frac{1-|{z}_{k}{|}^{2}}{1-|{\varphi }_{l}{\left(z\right)|}^{2}}<\epsilon ,$

whenever $\text{dist}\left(\varphi \left(z\right),\partial {U}^{n}\right)<\delta$. This is an extension of result by K. Madigan and A. Matheson [Trans. Am. Math. Soc. 347, 2679-2687 (1995; Zbl 0826.47023))], to $n\ge 1$.

##### MSC:
 47B33 Composition operators 32A18 Bloch functions, normal functions
##### Keywords:
Bloch space; polydisc; composition operator; Bergman metric
##### References:
 [1] Timoney, R., Bloch function in several complex variables, I, Bull. London Math. Soc., 1980, 12(37): 241. · Zbl 0428.32018 · doi:10.1112/blms/12.4.241 [2] Shi, J. H., Luo, L., Composition operators on the Bloch space of several complex variables, Acta Math. Sinica, 2000, 16 (1): 85. · Zbl 0967.32007 · doi:10.1007/s101149900028 [3] Madigan, K., Matheson, A., Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., 1995, 347 (7): 2679. · Zbl 0826.47023 · doi:10.2307/2154848