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ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

Part of: Holomorphic functions of several complex variables

Published online by Cambridge University Press:  12 December 2011

ZHANGJIAN HU
ZHANGJIAN HU*
Affiliation:
Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang, 313000, China (email: [email protected])
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Abstract

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Let Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which , where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by . In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

Keywords

Bergman spacesextended Cesàro operatorsessential norm

MSC classification

Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 2 , April 2012 , pp. 307 - 314
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[AC01]Aleman, A. and Cima, J., ‘An integral operator on H p and Hardy’s inequality’, J. Anal. Math. 85 (2001), 157–176.CrossRefGoogle Scholar
[AS95]Aleman, A. and Siskakis, A. G., ‘An integral operator on H p’, Complex Var. 28 (1995), 149–158.Google Scholar
[AS97]Aleman, A. and Siskakis, A. G., ‘Integration operators on Bergman spaces’, Indiana Univ. Math. J. 46 (1997), 337–356.CrossRefGoogle Scholar
[Hu03]Hu, Z. J., ‘Extended Cesàro operators on mixed norm spaces’, Proc. Amer. Math. Soc. 131 (2003), 2171–2179.CrossRefGoogle Scholar
[Hu04]Hu, Z. J., ‘Extended Cesàro operators on Bergman spaces’, J. Math. Anal. Appl. 296 (2004), 435–454.CrossRefGoogle Scholar
[HT10]Hu, Z. J. and Tang, X. M., ‘Schatten(-Herz) class extended Cesàro operators on Bergman spaces in the unit ball of ℂn’, Proc. Amer. Math. Soc. 138 (2010), 2803–2814.CrossRefGoogle Scholar
[Ra07]Rattya, J., ‘Integration operator acting on Hardy and weighted Bergman spaces’, Bull. Aust. Math. Soc. 75 (2007), 431–446.CrossRefGoogle Scholar
[Ru80]Rudin, W., Function Theory in the Unit Ball of ℂn (Springer, New York, 1980).CrossRefGoogle Scholar
[Zh05]Zhu, K. H., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226 (Springer, New York, 2005).Google Scholar