## Banach Journal of Mathematical Analysis

### Composition operators between generally weighted Bloch spaces and $Q_{log}^q$ space

#### Abstract

Let $\varphi$ be a holomorphic self-map of the open unit disk $D$ on the complex plane and $p,\ q>0.$ In this paper, the boundedness and compactness of composition operator $C_{\varphi}$ from generally weighted Bloch space $B^{p}_{\log}$ to $Q^{q}_{\log}$ are investigated.

#### Article information

Source
Banach J. Math. Anal. Volume 3, Number 1 (2009), 99-110.

Dates
First available in Project Euclid: 21 April 2009

http://projecteuclid.org/euclid.bjma/1240336427

Digital Object Identifier
doi:10.15352/bjma/1240336427

Mathematical Reviews number (MathSciNet)
MR2461750

Zentralblatt MATH identifier
1163.47019

#### Citation

Li, Haiying; Liu, Peide. Composition operators between generally weighted Bloch spaces and $Q_{log}^q$ space. Banach J. Math. Anal. 3 (2009), no. 1, 99--110. doi:10.15352/bjma/1240336427. http://projecteuclid.org/euclid.bjma/1240336427.

#### References

• K.R.M. Attele, Toeplitz and Hankel on Bergman one space , Hokaido, Math. J., 21 (1992), 279–293.
• R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15 (1995), 101–121.
• J. Cima and D. Stegenda, Hankel operators on $H^p$, in: Earl R. Berkson, N. T. Peck, J. Ulh(Eds.), Analysis at urbana 1, in: London Math. Soc. Lecture Note ser., Cambridge Univ. Press, Cambridge, 137 (1989), 133–150.
• C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Roton, 1995.
• P. Galanopoulos, On $B_\log$ to $Q_\log^p$ pullbacks, J. Math. Anal. Appl., 337(1) (2008), 712–725.
• H. Li, P. Liu and M. Wang, Composition operators between generally weighted Bloch spaces of polydisk, J. Inequal. Pure Appl. Math., 8(3) (2007), Article85, 1–8.
• K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., 347 (1995), 2679–2687.
• W. Ramey and D. Ulrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann., 291 (1991), 591–606.
• A. Siskakis and R. Zhao, A Volterra type operator on spaces on spaces of analytic functions, in: Contemp. Math., 232 (1999), 299–311.
• J. Xiao, The $Q_p$ corona theorem, Pacific J. Math., 194 (2000), 491–509.
• J. Xiao, Holomorphic $Q$ Classes, Lecture Notes in Math., Springer, >b<1767>/<, 2001.
• J. Xiao, Geometric $Q_p$ functions, Front. Math., Birkh$\ddota$user, 2006.
• R. Yoneda, The composition operators on the weighted Bloch space, Arch. Math., 78 ( 2002), 310–317.
• R. Zhao, On logarithmic Carleson measures, Acta Sci. Math.(Szeged), 69(3-4) (2003), 605–618.
• Kehe Zhu, Operator Theory on Function Spaces, New York, 1990.
• Kehe Zhu, Spaces of Holomorphic functions in the unit ball, New York, 2005.