Schatten class operators on the Bergman space over bounded symmetric domain
Abstract
Let $\Omega$ be a bounded symmetric domain in $\mathbb{C}^{n}$ with Bergman kernel $K(z, w)$. Let $dV_{\lambda}(z)=K(z, z)\frac{dV(z)}{C_{\lambda}}$, where $C_{\lambda}=\displaystyle\int_{\Omega}K(z, z)^{\lambda}dV(z)$, $\lambda\in\mathbb{R}$, $dV(z)$ is the volume measure of $\Omega$ normalized so that $K(z, 0)=K(0, w)=1$. In this paper we have shown that if the Toeplitz operator $T_{\phi}$ defined on $L_{a}^{2}(\Omega, \frac{dV}{C_{0}})$ belongs to the Schatten $p$-class, $1\leq p<\infty$, then $\widetilde{\phi}\in L^{p}(\Omega, d\eta)$, where $d\eta(z)=K(z, z)\frac{dV(z)}{C_{0}}$ and $\widetilde{\phi}$ is the Berezin transform of $\phi$. Further if $\phi\in L^{p}(\Omega, d\eta_{\lambda})$, then $\widetilde{\phi_{\lambda}}\in L^{p}(\Omega, d\eta_{\lambda})$ and $T_{\phi}^{\lambda}$ belongs to Schatten $p$-class. Here $d\eta_{\lambda}=K(z, z)\frac{dV(z)}{C_{\lambda}}$, the function $\widetilde{\phi_{\lambda}}$ is the Berezin transform of $\phi$ in $L_{a}^{2}(\Omega, dV_{\lambda})$ and $T_{\phi}^{\lambda}$ is the Toeplitz operator defined on $L_{a}^{2}(\Omega, dV_{\lambda})$. We also find conditions on bounded linear operator $C$ defined from $L_{a}^{2}(\Omega, dV_{\lambda})$ into itself such that $C$ belongs to the Schatten $p$-class by comparing it with positive Toeplitz operators defined on $L_{a}^{2}(\Omega, dV_{\lambda})$. Applications of these results are obtained and we also present Schatten class characterization of little Hankel operators defined on $L_{a}^{2}(\Omega, dV_{\lambda})$.
Key words: Bergman space, bounded symmetric domain, Toeplitz operators, little Hankel operators, Schatten class.
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