Schatten Class Weighted Composition Operators on the Bergman Space of the Unit Ball
Received:November 28, 2005  Revised:July 02, 2006
Key Words: Bergman space   weighted composition operator   Schatten class   Hilbert-Schmidt operator.
Fund Project:the National Natural Science Foundation of China (10471039), the Natural Science Foundation of the Education Commission of Jiangsu Province (03KJD140210, 06KJD110175) and the Natural Science Foundation of Xuzhou Institute of Technology (KY200508).
 Author Name Affiliation LIU Yong-min Department of Mathematics, Xuzhou Normal University, Jiangsu 221116, China YU Yan-yan Department of Computation Science, Xuzhou Institute of Technology, Jiangsu 221008, China
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We consider the Schatten class weighted composition operators on the Bergman space of the unit ball. The main result is several necessary and sufficient conditions for such kind of weighted composition operators belong to the Schatten-Von Neumann ideal $S_p$. As a corollary, we now have that $W_{\varphi, \psi}$ is a Hilbert-Schmidt operator if and only if $$\int_{B_n}\frac{|\psi(w)|^2}{(1-|\varphi(w)|^2)^{n+1}}\d V(w)<\infty.$$