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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 $$\phi=(\phi_1, \cdots, \phi_n)$$ 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 K. Madigan and A. Matheson [Trans. Am. Math. Soc. 347, 2679-2687 (1995; Zbl 0826.47023))], to $$n \geq 1$$.
Reviewer: Jinkee Lee (Pusan)

##### MSC:
 47B33 Linear composition operators 32A18 Bloch functions, normal functions of several complex variables
##### Keywords:
Bloch space; polydisc; composition operator; Bergman metric
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##### 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
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