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)


47B33 Linear composition operators
32A18 Bloch functions, normal functions of several complex variables
Full Text: DOI


[1] Timoney, R., Bloch function in several complex variables, I, Bull. London Math. Soc., 12, 37, 241-241 (1980) · Zbl 0416.32010 · 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, 16, 1, 85-85 (2000) · Zbl 0967.32007 · doi:10.1007/s101149900028
[3] Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., 347, 7, 2679-2679 (1995) · Zbl 0826.47023 · doi:10.2307/2154848
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