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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Equivalent norms on Fock spaces with some application to extended Cesaro operators


Author: Zhangjian Hu
Journal: Proc. Amer. Math. Soc. 141 (2013), 2829-2840
MSC (2010): Primary 32A37; Secondary 47B38
Published electronically: April 24, 2013
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Abstract: Let $ F_\gamma ^p$ be the Fock space of all holomorphic functions $ f$ in $ {\mathbf C}^n$ with the Fock norm

$\displaystyle \Vert f\Vert _{p, \gamma }=\left \{\int _{\mathbf {C}^n}\left \ve... ...mma \vert z\vert^2}{2}}\right \vert^{p}dA(z)\right \}^{\frac {1}{p}} <\infty , $

where $ p, \gamma $ are positive numbers. We prove that, given any positive integer $ m$, the Fock norm $ \Vert f\Vert _{p, \gamma }$ is equivalent to

$\displaystyle \sum _{\vert\alpha \vert\le m-1} \vert\partial ^\alpha f(0)\vert+... ...^ {-\frac {\gamma \vert z\vert^2}{2}} \right \vert^p dA(z)\right \}^{\frac 1p}.$

As some application we characterize these holomorphic functions $ g$ in $ {\mathbf C}^n$ for which the induced extended Cesaro operator $ T_g$ is bounded (or compact) from one Fock space $ F_\gamma ^p$ to another $ F_\gamma ^q$.

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Additional Information

Zhangjian Hu
Affiliation: Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang, 313000, People’s Republic of China
Email: huzj@hutc.zj.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11550-9
PII: S 0002-9939(2013)11550-9
Keywords: Fock spaces, extended Cesaro operators.
Received by editor(s): April 5, 2011
Received by editor(s) in revised form: November 9, 2011
Published electronically: April 24, 2013
Additional Notes: This research was partially supported by the National Natural Science Foundation of China (11271124, 11101139), the Natural Science Foundation of Zhejiang Province (Y6090036, Y6100219) and the Foundation of Creative Group in Universities of Zhejiang Province (T200924).
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society