zbMATH — the first resource for mathematics

Weighted composition operators from \(H^{\infty }\) to the Bloch space on the polydisc. (English) Zbl 1152.47016
Let \({\mathbb D}^n\) be the unit polydisc of \({\mathbb C}^n\), \(\varphi=(\varphi_1,\dots,\varphi_n)\) be a self-map of \({\mathbb D}^n\), and let \(\psi\) be holomorphic on \({\mathbb D}^n\). The main result of the present paper establishes that the weighted composition operator \(\psi C_\varphi\) is bounded from \(H^\infty({\mathbb D}^n)\) to \({\mathcal B}({\mathbb D}^n)\), i.e., from bounded functions to functions in the Bloch space, if and only if \(\psi \in {\mathcal B}({\mathbb D}^n)\) and
\[ \sup_{z\in {\mathbb D}^n} | \psi(z)| \sum_{k,j=1}^n \biggl|\frac{\partial \varphi_j}{\partial z_k}(z)\biggr| \frac{1-| z_k| ^2}{1-| \varphi_j(z)| ^2}<\infty. \] The corresponding compactness result replacing the supremum by “little oh” conditions is also shown.

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI EuDML
[1] R. M. Timoney, “Bloch functions in several complex variables. I,” The Bulletin of the London Mathematical Society, vol. 12, no. 4, pp. 241-267, 1980. · Zbl 0416.32010
[2] S. Ohno, “Weighted composition operators between H\infty and the Bloch space,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555-563, 2001. · Zbl 0997.47025
[3] S. Li and S. Stević, “Weighted composition operator between H\infty and \alpha -Bloch spaces in the unit ball,” to appear in Taiwanese Journal of Mathematics.
[4] J. Shi and L. Luo, “Composition operators on the Bloch space of several complex variables,” Acta Mathematica Sinica, vol. 16, no. 1, pp. 85-98, 2000. · Zbl 0967.32007
[5] S. Stević, “Composition operators between H\infty and a-Bloch spaces on the polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457-466, 2006. · Zbl 1118.47015
[6] Z. Zhou, “Composition operators on the Lipschitz space in polydiscs,” Science in China. Series A, vol. 46, no. 1, pp. 33-38, 2003. · Zbl 1217.32002
[7] Z. Zhou and J. Shi, “Compact composition operators on the Bloch space in polydiscs,” Science in China. Series A, vol. 44, no. 3, pp. 286-291, 2001. · Zbl 1024.47010
[8] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995. · Zbl 0873.47017
[9] S. Stević, “The generalized Libera transform on Hardy, Bergman and Bloch spaces on the unit polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 22, no. 1, pp. 179-186, 2003. · Zbl 1046.47026
[10] G. Benke and D.-C. Chang, “A note on weighted Bergman spaces and the Cesàro operator,” Nagoya Mathematical Journal, vol. 159, pp. 25-43, 2000. · Zbl 0981.32001
[11] S. Stević, “Cesàro averaging operators,” Mathematische Nachrichten, vol. 248-249, pp. 185-189, 2003. · Zbl 1024.47014
[12] S. Stević, “Hilbert operator on the polydisk,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 31, no. 2, pp. 135-142, 2003. · Zbl 1088.47024
[13] W. Rudin, Function Theory in the Unit Ball of \Bbb Cn, vol. 241 of Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1980. · Zbl 0495.32001
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.