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# 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.

### Article Information

 Title Schatten class operators on the Bergman space over bounded symmetric domain Source Methods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 193-212 MathSciNet MR3242704 zbMATH 1324.47052 Milestones Received 19/09/2012 Copyright The Author(s) 2014 (CC BY-SA)

### Authors Information

Namita Das P. G.
Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar, 751004, Odisha, India

School of Applied Sciences (Mathematics), KIIT University, Campus-3 (Kathajori Campus), Bhubaneswar, 751024, Odisha, India

### Citation Example

Namita Das and Madhusmita Sahoo, Schatten class operators on the Bergman space over bounded symmetric domain, Methods Funct. Anal. Topology 20 (2014), no. 3, 193-212.

### BibTex

@article {MFAT665,
AUTHOR = {Das, Namita and Sahoo, Madhusmita},
TITLE = {Schatten class operators on the Bergman space over bounded symmetric domain},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {20},
YEAR = {2014},
NUMBER = {3},
PAGES = {193-212},
ISSN = {1029-3531},
MRNUMBER = {MR3242704},
ZBLNUMBER = {1324.47052},
URL = {http://mfat.imath.kiev.ua/article/?id=665},
}