## Banach Journal of Mathematical Analysis

### New estimate of essential norm of composition followed by differentiation between Bloch-type spaces

#### Abstract

We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the $n$-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.

#### Article information

Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 118-137.

Dates
First available in Project Euclid: 14 October 2013

https://projecteuclid.org/euclid.bjma/1381782092

Digital Object Identifier
doi:10.15352/bjma/1381782092

Mathematical Reviews number (MathSciNet)
MR3161687

Zentralblatt MATH identifier
1323.47041

#### Citation

Liang, Yu-Xia; Zhou, Ze-Hua. New estimate of essential norm of composition followed by differentiation between Bloch-type spaces. Banach J. Math. Anal. 8 (2014), no. 1, 118--137. doi:10.15352/bjma/1381782092. https://projecteuclid.org/euclid.bjma/1381782092

#### References

• R. Aron and M. Lindström, Spectra of weighted composition operators on weighted Banach spaces of analytic functions, Israel J. of Math. 141 (2004), 263–276.
• R.E. Castillo, D.D. Clahane, J.F. Farías López and J.C. Ramos Fernández, Composition operators from logarithmic Bloch spaces to weighted Bloch spaces, Appl. Math. Comp. 219 (2013), 6692–6706.
• C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
• Z.S. Fang and Z.H. Zhou, Essential norms of composition operators between Bloch type spaces in the polydisk, Archiv der Mathematik 99 (2012), no. 6, 547–556.
• P. Gorkin and B.D. MacCluer, Essential norms of composition operators, Integral Equations Operator Theory 48 (2004), 27–40.
• O. Hyvärinen and M. Lindström, Estimates of essential norms of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl. 393 (2012), 38–44.
• O. Hyvärinen, M. Kemppainen, M. Lindström, A. Rautio, and E. Saukko, The essential norm of weighted composition operators on weighted Banach spaces of analytic functions, Integral Equations Operator Theory 72 (2012), 151–157.
• S. Li and S. Stević, Composition followed by differentiation between Bloch type spaces, J. Comput. Anal. Appl. 9 (2007), no. 2, 195–206.
• B.D. MacCluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437–1458.
• J.S. Manhas and R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl. 389 (2012), 32–47.
• J.C. Ramos-Fernández, Composition operators on Bloch-Orlicz type spaces, Appl. Math. Comp. 217 (2010), 3392–3402.
• J.H. Shapiro,Composition operators and Classical Function Theory, Springer-Verlag, New York, 1993.
• S. Stević, Characterizations of composition followed by differentiation between Bloch-type spaces, Appl. Math. Comput. 218 (2011), 4312–4316.
• Y. Wu and H. Wulan,Products of differentiation and composition operators on the Bloch space, Collect. Math. 63 (2012), 93–107.
• H. Wulan, D. Zheng, and K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), 3861–3868.
• R. Zhao, Essential norms of composisition operators between Bloch type spces, Proc. Amer. Math. Soc. 138 (2010), 2537–2546.
• Z.H. Zhou and R.Y. Chen,Weighted composition operators fom $F(p, q, s)$ to Bloch type spaces, Internat. J. Math. 19 (2008), no. 8, 899–926.
• Z.H. Zhou and J.H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J. 50 (2002), 381–405.
• L. Zhang and Z.H. Zhou, Hilbert-Schmidt differences of composition operators between the weighted Bergman spaces on the unit ball, Banach J. Math. Anal. 7 (2013), 160–172.
• H.G. Zeng and Z.H. Zhou, Essential norm estimate of a composition operator between Bloch-type spaces in the unit ball, Rocky Mountain J. Math. 42 (2012), 1049–1071.
• K. Zhu, Operator Theory in Function Spces, Marcel Dekker, New YOrk, 1990.