Composition operators between \(H^\infty\) and \(a\)-Bloch spaces on the polydisc. (English) Zbl 1118.47015

Let \(U^n\) be the unit polydisc of \({\mathbb C}^n\), \(\alpha>0\), and let \(H^\infty(U^n)\) and \({\mathcal B}^\alpha(U^n)\) denote the space of holomorphic functions in the unit disc on \(U^n\) which are bounded and belong to \(\alpha\)-Bloch space, i.e., \(\sup_{| z_k| <1}\sum_{k=1}^n(1-| z_k| ^2)^\alpha | \frac{\partial f}{\partial z_k}(z)| <\infty\), respectively. Given a holomorphic self-map on \(U^n\), \(\phi=(\phi_1,\dots,\phi_n)\), the author shows that for \(\alpha\geq 1\) the composition operator \(C_\phi:H^\infty(U^n)\to {\mathcal B}^\alpha(U^n)\), defined by \(C_\phi(f)=f\circ \phi\), is compact if and only if for every \(\varepsilon >0\), there exists \(0<\delta<1\) such that if dist\((\phi(z),\partial U^n)<\delta\), then
\[ \sum_{k,l=1}^n\frac{(1-| z_k| ^2)^\alpha}{1-| \phi_l(z)| ^2} \biggl| \frac{\partial \phi_l}{\partial z_k}(z)\biggr| <\varepsilon. \] He uses his result to produce noncompact composition operators form \(H^\infty(U^n)\) to \({\mathcal B}^1(U^n)\).


47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
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[1] G. Benke and Chang, D. C., A note on weighted Bergman spaces and the Ces‘aro operator. Nagoya Math. J. 159 (2000), 25 - 43. · Zbl 0981.32001
[2] Stević, S., Ces‘aro averaging operators. Math. Nachr. 248-249 (2003), 185 - 189. · Zbl 1024.47014
[3] Stević, S., Hilbert operator on the polydisk. Bull. Inst. Math. Acad. Sinica 31 (2003)(2), 135 - 142. · Zbl 1088.47024
[4] Stević, S., The generalized Libera transform on Hardy, Bergman and Bloch spaces on the unit polydisc. Z. Anal. Anwendungen 22 (2003)(1), 179 - 186. · Zbl 1046.47026
[5] Zhu, K., The Bergman spaces, the Bloch spaces, and Gleason’s problem. Trans. Amer. Math. Soc. 309 (1988)(1), 253 - 268. · Zbl 0657.32002
[6] Zhu, K., Duality and Hankel operators on the Bergman spaces of bounded symmetric domains. J. Funct. Anal. 81 (1988), 260 - 278. · Zbl 0669.47019
[7] Zhou, Z. H. and Shi, J. H., Compact composition operators on the Bloch space in polydiscs. Sci. China Ser. A 44 (2001), 286 - 291. · Zbl 1024.47010
[8] Zhou, Z. H. and Shi, J. H., Composition operators on the Bloch space in polydiscs. Complex Variables 46 (2001)(1), 73 - 88. · Zbl 1026.47018
[9] Zhou, Z. H. and Shi, J. H., Compactness of composition operators on the Bloch space in classical bounded symmetric domains. Michigan Math. J. 50 (2002)(2), 381 - 405. · Zbl 1044.47021
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