×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.