# NEW CHARACTERIZATIONS OF COMPOSITION OPERATORS BETWEEN BLOCH TYPE SPACES IN THE UNIT BALL

• Fang, Zhong-Shan ;
• Zhou, Ze-Hua
• Published : 2015.05.31
• 47 6

#### Abstract

In this paper, we give new characterizations of the boundedness and compactness of composition operators $C_{\varphi}$ between Bloch type spaces in the unit ball $\mathbb{B}^n$, in terms of the power of the components of ${\varphi}$, where ${\varphi}$ is a holomorphic self-map of $\mathbb{B}^n$.

#### Keywords

composition operator;compactness;Bloch type spaces;unit ball;several complex variables

#### References

1. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
2. H. Chen and P. Gauthier, Composition operators on $\mu$-Bloch spaces, Canad. J. Math. 61 (2009), no. 1, 50-75. https://doi.org/10.4153/CJM-2009-003-1
3. J. Dai, Compact composition operators on the Bloch space of the unit ball, J. Math. Anal. Appl. 386 (2012), no. 1, 294-299. https://doi.org/10.1016/j.jmaa.2011.07.067
4. K. Esmaeili and M. Lindstrom, Weighted composition operators between Zygmund type spaces and their essential norms, Integral Equations Operator Theory 75 (2013), no. 4, 473-490. https://doi.org/10.1007/s00020-013-2038-4
5. Z. S. Fang, Compact composition operators between Bloch type spaces in the polydisk, Abstr. Appl. Anal. 2012 (2012), Article ID 732709, 7 pp.
6. Z. S. Fang and Z. H. Zhou, Essential norms of composition operators between Bloch type spaces in the polydisk, Arch. Math. (Basel) 99 (2012), no. 6, 547-556. https://doi.org/10.1007/s00013-012-0457-0
7. O. Hyvarinen, M. Kemppainen, M. Lindstrom, 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), no. 2, 151-157. https://doi.org/10.1007/s00020-011-1919-7
8. O. Hyvarinen and M. Lindstrom, Estimates of essential norms of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl. 393 (2012), no. 1, 38-44. https://doi.org/10.1016/j.jmaa.2012.03.059
9. Y. X. Liang and Z. H. Zhou, Essential norm of the product of differentiation and com- position operators between Bloch-type spaces, Arch. Math. (Basel) 100 (2013), no. 4, 347-360. https://doi.org/10.1007/s00013-013-0499-y
10. Y. X. Liang and Z. H. Zhou, New estimate of essential norm of composition followed by differentiation between Bloch-type spces, Banach J. Math. Anal. 7 (2013), 160-172. https://doi.org/10.15352/bjma/1358864556
11. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679-2687. https://doi.org/10.1090/S0002-9947-1995-1273508-X
12. J. Manhas and R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl. 389 (2012), no. 1, 32-47. https://doi.org/10.1016/j.jmaa.2011.11.039
13. J. H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), no. 2, 375-404. https://doi.org/10.2307/1971314
14. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.
15. S. Stevic, On new Bloch-type spaces, Appl. Math. Comput. 215 (2009), no. 2, 841-849. https://doi.org/10.1016/j.amc.2009.06.009
16. S. Stevic, R. Y. Chen, and Z. H. Zhou, Weighted composition operators between Bloch-type spaces in the polydisk, Sbornik Mathematics 201 (2010), no. 2, 289-319. https://doi.org/10.1070/SM2010v201n02ABEH004073
17. Y. Wu and H. Wulan, Products of differentiation and composition operators on the Bloch space, Collect. Math. 63 (2012), no. 1, 93-107. https://doi.org/10.1007/s13348-010-0030-8
18. H. Wulan, D. Zheng, and K. Zhu, Compact composition operators on BMO and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3861-3868. https://doi.org/10.1090/S0002-9939-09-09961-4
19. R. Zhao, Essential norms of composition operators between Bloch type spaces, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2537-2546. https://doi.org/10.1090/S0002-9939-10-10285-8
20. Z. H. Zhou, Y. X. Liang, and X. T. Dong, Weighted composition operator between weighted-type space and Hardy space on the unit ball, Ann. Polon. Math. 104 (2012), no. 3, 309-319. https://doi.org/10.4064/ap104-3-7
21. Z. H. Zhou and J. H. Shi, Compact composition operators on the Bloch space in the polydisk, Sci. China Ser. A 44 (2001), no. 3, 286-291. https://doi.org/10.1007/BF02878708
22. Z. H. Zhou and J. H. Shi, Composition operators on the Bloch space in the polydisk, Complex Variables 46 (2001), no. 1, 73-88. https://doi.org/10.1080/17476930108815398
23. 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), no. 2, 381-405. https://doi.org/10.1307/mmj/1028575740
24. 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), no. 3, 1049-1071. https://doi.org/10.1216/RMJ-2012-42-3-1049
25. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

#### Acknowledgement

Supported by : National Natural Science Foundation of China