## Abstract and Applied Analysis

### Essential Norms of Weighted Composition Operators from the $\alpha$-Bloch Space to a Weighted-Type Space on the Unit Ball

Stevo Stević

#### Abstract

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from $\alpha$-Bloch spaces to the weighted-type space ${H}_{\mu }^{\infty }$ on the unit ball for the case $\alpha \geq 1$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 279691, 11 pages.

Dates
First available in Project Euclid: 10 February 2009

https://projecteuclid.org/euclid.aaa/1234298995

Digital Object Identifier
doi:10.1155/2008/279691

Mathematical Reviews number (MathSciNet)
MR2453144

Zentralblatt MATH identifier
1160.32011

#### Citation

Stević, Stevo. Essential Norms of Weighted Composition Operators from the $\alpha$ -Bloch Space to a Weighted-Type Space on the Unit Ball. Abstr. Appl. Anal. 2008 (2008), Article ID 279691, 11 pages. doi:10.1155/2008/279691. https://projecteuclid.org/euclid.aaa/1234298995

#### References

• 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, 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 B, vol. 22, no. 1, pp. 61--68, 2007.
• 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.
• 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.
• K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
• X. Fu and X. Zhu, Weighted composition operators on some weighted spaces in the unit ball,'' Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.
• P. Gorkin and B. D. MacCluer, Essential norms of composition operators,'' Integral Equations and Operator Theory, vol. 48, no. 1, pp. 27--40, 2004.
• 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, 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. 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, 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, Weighted composition operators between $H^\infty$ and $\alpha$-Bloch space čommentComment on ref. [11?]: Please update the information of this reference, if possible. in the unit ball,'' to appear in Taiwanese Journal of Mathematics.
• L. Luo and S.-I. Ueki, Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of $\mathbbC^n$,'' Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 88--100, 2007.
• B. D. MacCluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces,'' The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437--1458, 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. Ohno, Weighted composition operators between $H^\infty$ and the Bloch space,'' Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555--563, 2001.
• 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.
• 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.
• 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. 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.
• 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.
• 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.
• X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces,'' Integral Transforms and Special Functions, vol. 18, no. 3, pp. 223--231, 2007.
• S. Stević, On an integral operator on the unit ball in $\mathbbC^n$,'' Journal of Inequalities and Applications, vol. 2005, no. 1, pp. 81--88, 2005.
• 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. 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. \endthebibliography