Abstract and Applied Analysis

Weighted Composition Operators on Some Weighted Spaces in the Unit Ball

Xiaohong Fu and Xiangling Zhu

Full-text: Open access

Abstract

Let B n be the unit ball of C n , H ( B n ) the space of all holomorphic functions in B n . Let u H ( B n ) and α be a holomorphic self-map of B n . For f H ( B n ) , the weigthed composition operator u C α is defined by ( u C α f ) ( z ) = u ( z ) f ( α ( z ) ) , z B n . The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 605807, 8 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969177

Digital Object Identifier
doi:10.1155/2008/605807

Mathematical Reviews number (MathSciNet)
MR2417227

Zentralblatt MATH identifier
1160.47024

Citation

Fu, Xiaohong; Zhu, Xiangling. Weighted Composition Operators on Some Weighted Spaces in the Unit Ball. Abstr. Appl. Anal. 2008 (2008), Article ID 605807, 8 pages. doi:10.1155/2008/605807. https://projecteuclid.org/euclid.aaa/1220969177


Export citation

References

  • Z. Hu, ``Extended Cesàro operators on mixed norm spaces,'' Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171--2179, 2003.
  • 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.
  • 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.
  • 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.
  • S. Li, ``Fractional derivatives of Bloch-type functions,'' Siberian Mathematical Journal, vol. 46, no. 2, pp. 308--314, 2005.
  • S. Li, ``Derivative-free characterizations of Bloch spaces,'' Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 253--258, 2008.
  • S. Li and S. Stević, ``Riemann-Stieltjes-type integral operators on the unit ball in $\mathbbC^n$,'' Complex Variables and Elliptic Equations, vol. 52, no. 6, pp. 495--517, 2007.
  • S. Li and S. Stević, ``Some characterizations of the čommentComment on ref. [16?]: Please update the information of this reference, if possible. Besov space and the $\alpha $-Bloch space,'' Journal of Mathematical Analysis and Applications. In press.
  • 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.
  • S. Stević, ``On an integral operator on the unit ball in $\mathbbC^n$,'' Journal of Inequalities and Applications, no. 1, pp. 81--88, 2005.
  • S. Stević, ``On Bloch-type functions with Hadamard gaps,'' Abstract and Applied Analysis, vol. 2007, Article ID 39176, 8 pages, 2007.
  • 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.
  • K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
  • D. Girela, J. Peláez, F. Pérez-González, and J. Rättyä, ``Carleson measures čommentComment on ref. [3?]: Please update the information of this reference, if possible. for the Bloch space,'' preprint, 2008.
  • 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.
  • X. Zhu, ``Weighted composition operators between $H^\infty $ and Bergman type spaces,'' Korean Mathematical Society, vol. 21, no. 4, pp. 719--727, 2006.
  • 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.
  • 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.
  • S. Li and S. Stević, ``Weighted composition operators between $H^\infty $ and $\alpha $-Bloch space in the čommentComment on ref. [17?]: Please update the information of this reference, if possible. unit ball,'' to appear in Taiwanese Journal of Mathematics.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.