## Abstract and Applied Analysis

### Weighted Composition Operators from $F(p,q,s)$ Spaces to ${H}_{\mu }^{\infty }$ Spaces

Xiangling Zhu

#### Abstract

Let $H(B)$ denote the space of all holomorphic functions on the unit ball $B$. Let $u\in H(B)$ and $\varphi$ be a holomorphic self-map of $B$. In this paper, we investigate the boundedness and compactness of the weighted composition operator $u{C}_{\varphi }$ from the general function space $F(p,q,s)$ to the weighted-type space ${H}_{\mu }^{\infty }$ in the unit ball.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 290978, 14 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745548

Digital Object Identifier
doi:10.1155/2009/290978

Mathematical Reviews number (MathSciNet)
MR2501015

Zentralblatt MATH identifier
1173.30037

#### Citation

Zhu, Xiangling. Weighted Composition Operators from $F(p,q,s)$ Spaces to ${H}_{\mu }^{\infty }$ Spaces. Abstr. Appl. Anal. 2009 (2009), Article ID 290978, 14 pages. doi:10.1155/2009/290978. https://projecteuclid.org/euclid.aaa/1268745548

#### 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.
• 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.
• 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. Stević, On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,'' Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008.
• K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
• 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 and S. Stević, Some characterizations of the Besov space and the $\alpha$-Bloch space,'' Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 262--273, 2008.
• 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.
• C. Ouyang, W. Yang, and R. Zhao, Möbius invariant $Q_p$ spaces associated with the Green's function on the unit ball of $C^n$,'' Pacific Journal of Mathematics, vol. 182, no. 1, pp. 69--99, 1998.
• S. Li, Riemann-Stieltjes operators from $F(p,q,s)$ spaces to $\alpha$-Bloch spaces on the unit ball,'' Journal of Inequalities and Applications, vol. 2006, Article ID 27874, 14 pages, 2006.
• S. Li and S. Stević, Compactness of Riemann-Stieltjes operators between $F(p,q,s)$ spaces and $\alpha$-Bloch spaces,'' Publicationes Mathematicae Debrecen, vol. 72, no. 1-2, pp. 111--128, 2008.
• X. J. Zhang, Multipliers on some holomorphic function spaces,'' Chinese Annals of Mathematics. Series A, vol. 26, no. 4, pp. 477--486, 2005.
• R. Zhao, On a general family of function spaces,'' Annales Academiæ Scientiarum Fennicæ. Series A I. Mathematica. Dissertationes, no. 105, p. 56, 1996.
• 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,'' Communications of the 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.
• S. Li and S. Stević, Weighted composition operators between $H^\infty$ and $\alpha$-Bloch spaces in the unit ball,'' Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1625--1639, 2008.
• 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. 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.
• L. Jiang and C. Ouyang, Cyclic behavior of linear fractional composition operators in the unit ball of $\mathbbC^n$,'' Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 601--612, 2008.
• S. Li and S. Stević, Composition followed by differentiation from mixed norm spaces to $\alpha$-Bloch spaces,'' Matematicheskij Sbornik, vol. 199, no. 12, pp. 117--128, 2008.
• S. Li and S. Stević, Products of Volterra type operator and composition operator from $H^\infty$ and Bloch spaces to Zygmund spaces,'' Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 40--52, 2008.
• S. Li and S. Stević, Weighted composition operators from Zygmund spaces into Bloch spaces,'' Applied Mathematics and Computation, vol. 206, no. 2, pp. 825--831, 2008.
• 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.
• 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.
• S. Stević, Essential norms of weighted composition operators from the $\alpha$-Bloch space to a weighted-type space on the unit ball,'' Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008.
• S. Stević, Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,'' Utilitas Mathematica, vol. 76, pp. 59--64, 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.
• 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, On an integral operator on the unit ball in $\mathbbC^n$,'' Journal of Inequalities and Applications, no. 1, pp. 81--88, 2005.
• 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.
• 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 in a mixed norm space on the polydisk,'' Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 48, no. 3, pp. 694--706, 2007.