IJPAM: Volume 78, No. 2 (2012)

WEIGHTED COMPOSITION OPERATORS ON
WEIGHTED BERGMAN SPACES OF THE UNIT BALL

Waleed Al-Rawashdeh$^1$, Sivaram K. Narayan$^2$
$^1$Department of Mathematical Sciences
Montana Tech of The University of Montana
1300, West Park Street, Butte, Montana, 59701, USA
$^2$Department of Mathematics
Central Michigan University
Washington Street, Mount Pleasant, Michigan, 48859, USA


Abstract. Suppose $\varphi$ is an analytic self map of $\mathbb{B}_n$ and $\psi$ is analytic on $\mathbb{B}_n$. Then a weighted composition operator induced by $\varphi$ with weight $\psi$ is given by $\left(W_{\psi, \varphi}f\right)\left(z\right) = \psi(z) f(\varphi(z))$ for $z$ in $\mathbb{B}_n$ and f analytic on $\mathbb{B}_n$. Given $W_{\psi, \varphi} : A^{p}_{\alpha}(\mathbb{B}_{n}) \rightarrow A^{q}_{\beta}(\mathbb{B}_{n})$ we characterize boundedness and compactness of $W_{\psi, \varphi}$, where $0<p, q<\infty$ and $-1<\alpha, \beta<\infty$. We also characterize the Schatten $p$-class weighted composition operators $S_{p}\left(A^{2}_{\alpha}(\mathbb{B}_{n})\right)$ for $0<p<\infty$ and $-1<\alpha<\infty$.

Received: December 27, 2011

AMS Subject Classification: 47B38, 32A35, 32A36

Key Words and Phrases: weighted composition operators, Bergman spaces, Toeplitz operators, the Berezin transform, Schatten $p$-class, unit ball of $\mathbb{C}^n$

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 2