##
**Weighted composition operators on some weighted spaces in the unit ball.**
*(English)*
Zbl 1160.47024

For a positive weight function \(\mu\) on \([0,1)\), the space \(H^\infty_\mu\) consists of analytic functions \(f\) in the unit disc \(D\) such that
\[
\sup_{z\in D}\mu(| z| )| f(z)| <\infty.
\]
The paper studies weighted composition operators \(uC_\varphi\) on the spaces \(H^\infty_\mu\), where \(u\) is an analytic function in \(D\) and \(\varphi\) is an analytic self-map of \(D\). The main problems studied include the boundedness and compactness of these operators on various \(H^\infty_\mu\) spaces.

Reviewer: Kehe Zhu (Albany)

### MSC:

47B33 | Linear composition operators |

PDF
BibTeX
XML
Cite

\textit{X. Fu} and \textit{X. Zhu}, Abstr. Appl. Anal. 2008, Article ID 605807, 8 p. (2008; Zbl 1160.47024)

### References:

[1] | Z. Hu, “Extended Cesàro operators on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171-2179, 2003. · Zbl 1054.47023 |

[2] | A. L. Shields and D. L. Williams, “Bonded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287-302, 1971. · Zbl 0227.46034 |

[3] | S. Stević, “Norm of weighted composition operators from Bloch space to H\mu \infty on the unit ball,” Ars Combinatoria, vol. 88, pp. 125-127, 2008. · Zbl 1224.30195 |

[4] | D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006. · Zbl 1131.47018 |

[5] | S. Li, “Fractional derivatives of Bloch-type functions,” Siberian Mathematical Journal, vol. 46, no. 2, pp. 308-314, 2005. · Zbl 1102.32003 |

[6] | S. Li, “Derivative-free characterizations of Bloch spaces,” Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 253-258, 2008. · Zbl 1138.32001 |

[7] | S. Li and S. Stević, “Riemann-Stieltjes-type integral operators on the unit ball in \Bbb Cn,” Complex Variables and Elliptic Equations, vol. 52, no. 6, pp. 495-517, 2007. · Zbl 1124.47022 |

[8] | S. Li and S. Stević, “Some characterizations of the Besov space and the \alpha -Bloch space,” Journal of Mathematical Analysis and Applications. In press. · Zbl 1156.32002 |

[9] | S. Li and H. Wulan, “Characterizations of \alpha -Bloch spaces on the unit ball,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 58-63, 2008. · Zbl 1204.32006 |

[10] | S. Stević, “On an integral operator on the unit ball in \Bbb Cn,” Journal of Inequalities and Applications, no. 1, pp. 81-88, 2005. · Zbl 1074.47013 |

[11] | S. Stević, “On Bloch-type functions with Hadamard gaps,” Abstract and Applied Analysis, vol. 2007, Article ID 39176, 8 pages, 2007. · Zbl 1157.32301 |

[12] | 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 |

[13] | K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005. · Zbl 1067.32005 |

[14] | D. Girela, J. Peláez, F. Pérez-González, and J. Rättyä, “Carleson measures for the Bloch space,” preprint, 2008. · Zbl 1185.30056 |

[15] | 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 |

[16] | X. Zhu, “Weighted composition operators between H\infty and Bergman type spaces,” Korean Mathematical Society, vol. 21, no. 4, pp. 719-727, 2006. · Zbl 1160.47028 |

[17] | S. Stević, “Weighted composition operators between mixed norm spaces and H\alpha \infty spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007. · Zbl 1138.47019 |

[18] | X. Tang, “Weighted composition operators between Bers-type spaces and Bergman spaces,” Applied Mathematics: A Journal of Chinese Universities, vol. 22, no. 1, pp. 61-68, 2007. · Zbl 1125.47017 |

[19] | S. Li and S. Stević, “Weighted composition operators between H\infty and \alpha -Bloch space in the unit ball,” to appear in Taiwanese Journal of Mathematics. · Zbl 1177.47032 |

[20] | S. Li and S. Stević, “Weighted composition operators from H\infty to the Bloch space on the polydisc,” Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007. · Zbl 1130.47015 |

[21] | S. Li and S. Stević, “Weighted composition operators from \alpha -Bloch space to H\infty on the polydisc,” Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911-925, 2007. · Zbl 1130.47015 |

[22] | S. Stević, “Composition operators between H\infty and \alpha -Bloch spaces on the polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457-466, 2006. · Zbl 1118.47015 |

[23] | S. Li and S. Stević, “Composition followed by differentiation between Bloch type spaces,” Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195-205, 2007. · Zbl 1132.47026 |

[24] | S. Li and S. Stević, “Weighted composition operators from Bergman-type spaces into Bloch spaces,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 3, pp. 371-385, 2007. · Zbl 1130.47016 |

[25] | S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282-1295, 2008. · Zbl 1135.47021 |

[26] | S. Li and S. Stević, “Products of composition and integral type operators from H\infty to the Bloch space,” Complex Variables and Elliptic Equations, vol. 53, no. 5, pp. 463-474, 2008. · Zbl 1159.47019 |

[27] | S. Li and S. Stević, “Products of Volterra type operator and composition operator from H\infty and Bloch spaces to the Zygmund space,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 40-52, 2008. · Zbl 1145.47022 |

[28] | S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191-215, 2003. · Zbl 1042.47018 |

[29] | A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872-884, 2000. · Zbl 0959.47016 |

[30] | S.-I. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008. · Zbl 1167.47020 |

[31] | S. Ye, “Weighted composition operator between the little \alpha -Bloch spaces and the logarithmic Bloch,” Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 243-252, 2008. · Zbl 1152.47019 |

[32] | X. Zhu, “Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces,” Indian Journal of Mathematics, vol. 49, no. 2, pp. 139-150, 2007. · Zbl 1130.47017 |

[33] | S. Stević, “Boundedness and compactness of an integral operator on a weighted space on the polydisc,” Indian Journal of Pure and Applied Mathematics, vol. 37, no. 6, pp. 343-355, 2006. · Zbl 1121.47032 |

[34] | S. Stević, “Boundedness and compactness of an integral operator in a mixed norm space on the polydisk,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 48, no. 3, pp. 694-706, 2007. · Zbl 1164.47331 |

[35] | K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679-2687, 1995. · Zbl 0826.47023 |

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.