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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.


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Article Information

TitleSchatten class operators on the Bergman space over bounded symmetric domain
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 193-212
MathSciNet MR3242704
zbMATH 1324.47052
MilestonesReceived 19/09/2012
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

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

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


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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},
}


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