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Compact composition operators on the Bloch space in polydiscs. (English) Zbl 1024.47010
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 $n \geq 1$.

47B33Composition operators
32A18Bloch functions, normal functions
Full Text: DOI
[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