## Abstract and Applied Analysis

### Generalized Composition Operators from ${\scr B}_{\mu }$ Spaces to ${Q}_{K,\omega }(p,q)$ Spaces

#### Abstract

Let $0, let $-2, and let $\phi$ be an analytic self-map of $\mathrm{\Bbb D}$ and $g\in H(\mathrm{\Bbb D})$. The boundedness and compactness of generalized composition operators $({C}_{\phi }^{g}f)(z)={\int }_{0}^{z}\mathrm{‍}{f}^{\mathrm{\text{'}}}(\phi (\xi ))g(\xi )d\xi , z\in \mathrm{\Bbb D}, f\in H(\mathrm{\Bbb D})$, from ${\scr B}_{\mu }$ (${\scr B}_{\mu ,0}$) spaces to ${Q}_{K,\omega }(p,q)$ spaces are investigated.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 897389, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278797

Digital Object Identifier
doi:10.1155/2014/897389

Mathematical Reviews number (MathSciNet)
MR3193557

#### Citation

Li, Haiying; Ma, Tianshui. Generalized Composition Operators from ${\scr B}_{\mu }$ Spaces to ${Q}_{K,\omega }(p,q)$ Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 897389, 6 pages. doi:10.1155/2014/897389. https://projecteuclid.org/euclid.aaa/1412278797

#### References

• 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.
• R. A. Rashwan, A. El-Sayed Ahmed, and A. Kamal, “Integral characterizations of weighted Bloch spaces and ${Q}_{K,\omega }(p,q)$ spaces,” Mathematica, vol. 51, no. 74, pp. 63–76, 2009.
• F. Zhang and Y. Liu, “Generalized composition operators from Bloch type spaces to ${Q}_{K}$ type spaces,” Journal of Function Spaces and Applications, vol. 8, no. 1, pp. 55–66, 2010.
• C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, Fla, USA, 1995.
• P. Galanopoulos, “On ${B}_{\log }$ to ${Q}_{\log }^{p}$ pullbacks,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 712–725, 2008.
• M. Kotilainen, “On composition operators in ${Q}_{K}$ type spaces,” Journal of Function Spaces and Applications, vol. 5, no. 2, pp. 103–122, 2007.
• H. Li and P. Liu, “Composition operators between generally weighted Bloch space and ${Q}_{\text{l}\text{o}\text{g}}^{q}$ space,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 99–110, 2009.
• B. D. MacCluer and J. H. Shapiro, “Angular derivatives and compact composition operators on the Hardy and Bergman spaces,” Canadian Journal of Mathematics, vol. 38, no. 4, pp. 878–906, 1986.
• C. Yang, W. Xu, and M. Kotilainen, “Composition operators from Bloch type spaces into ${Q}_{K}$ type spaces,” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 26–34, 2011.
• R. Yoneda, “The composition operators on weighted Bloch space,” Archiv der Mathematik, vol. 78, no. 4, pp. 310–317, 2002.
• J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993.
• X. Zhu, “Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces,” Journal of the Korean Mathematical Society, vol. 46, no. 6, pp. 1219–1232, 2009.
• 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.
• A. E.-S. Ahmed and A. Kamal, “Generalized composition operators on ${Q}_{K,\omega }(p,q)$ spaces,” Mathematical Sciences, vol. 6, article 14, 9 pages, 2012.
• S. Rezaei and H. Mahyar, “Generalized composition operators from logarithmic Bloch type spaces to ${Q}_{K}$ type spaces,” Mathematics Scientific Journal, vol. 8, no. 1, pp. 45–57, 2012.
• Sh. Rezaei and H. Mahyar, “Essential norm of generalized composition operators from weighted Dirichlet or Bloch type spaces to ${Q}_{K}$ type spaces,” Iranian Mathematical Society. Bulletin, vol. 39, no. 1, pp. 151–164, 2013.
• W. Yang and X. Meng, “Generalized composition operators from $F(p,q,s)$ spaces to Bloch-type spaces,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2513–2519, 2010.
• S. Stević, “Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces,” Utilitas Mathematica, vol. 77, pp. 167–172, 2008.
• L. Zhang and Z.-H. Zhou, “Generalized composition operator from Bloch-type spaces to mixed-norm space on the unit ball,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 523–532, 2012.
• X. Zhu, “Generalized composition operators and Volterra composition operators on Bloch spaces in the unit ball,” Complex Variables and Elliptic Equations, vol. 54, no. 2, pp. 95–102, 2009. \endinput