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ESSENTIAL COMMUTATIVITY OF SOME INTEGRAL AND COMPOSITION OPERATORS

Published online by Cambridge University Press:  20 October 2011

ZE-HUA ZHOU*
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, PR China (email: [email protected])
LIANG ZHANG
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, PR China (email: [email protected])
HONG-GANG ZENG
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, PR China (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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In general, multiplication of operators is not essentially commutative in an algebra generated by integral-type operators and composition operators. In this paper, we characterize the essential commutativity of the integral operators and composition operators from a mixed-norm space to a Bloch-type space, and give a complete description of the universal set of integral operators. Corresponding results for boundedness and compactness are also obtained.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first author was supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 10671141).

References

[1]Aleman, A. and Cima, J. A., ‘An integral operator on H p and Hardy’s inequality’, J. Anal. Math. 85 (2001), 157176.CrossRefGoogle Scholar
[2]Aleman, A. and Siskakis, A. G., ‘Integration operators on Bergman spaces’, Indiana Univ. Math. 46(2) (1997), 337356.CrossRefGoogle Scholar
[3]Cowen, C. C. and MacCluer, B., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).Google Scholar
[4]Li, S. and Stević, S., ‘Composition followed by differentiation between Bloch-type spaces’, J. Comput. Anal. Appl. 9(2) (2007), 195205.Google Scholar
[5]Li, S. X. and Stević, S., ‘Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces’, Math. Sbornik 199(12) (2008), 117128.CrossRefGoogle Scholar
[6]Siskakis, A. G. and Zhao, R. H., ‘A Volterra type operator on spaces of analytic functions. Function spaces (Edwardsville, IL, 1998)’, Contemp. Math. 232 (1999), 299311.CrossRefGoogle Scholar
[7]Tjani, M., ‘Compact composition operators on Besov spaces’, Trans. Amer. Math. Soc. 355(11) (2003), 46834698.Google Scholar
[8]Wu, Z., ‘Carleson measures and multipliers for Dirichlet spaces’, J. Funct. Anal. 169 (1999), 148163.CrossRefGoogle Scholar
[9]Wulan, H., Zheng, D. and Zhu, K., ‘Compact composition operators on BMOA and the Bloch space’, Proc. Amer. Math. Soc. 137 (2009), 38613868.CrossRefGoogle Scholar
[10]Xiao, J., ‘The Q p Carelson measure problem’, Adv. in Math. 217 (2008), 20752088.CrossRefGoogle Scholar
[11]Zhou, Z. H., ‘Composition operator on the Lipschitz space in polydiscs’, Sci. China Ser. A 46(1) (2003), 3338.Google Scholar
[12]Zhou, Z. H. and Chen, R. Y., ‘Weighted composition operators fom F(p,q,s) to Bloch type spaces’, Internat. J. Math. 19(8) (2008), 899926.Google Scholar
[13]Zhou, Z. H. and Shi, J. H., ‘Compactness of composition operators on the Bloch space in classical bounded symmetric domains’, Michigan Math. J. 50 (2002), 381405.Google Scholar
[14]Zhu, K. H., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226 (Springer, New York, 2005).Google Scholar