M. Sahoo

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 $L_{a}^{2}(\Omega, dV_{\lambda})$.

Let $\mathbb{D}=\{{z\in\mathbb{C}:|z|<1}\}$ and $L^2_a(\mathbb{D})$ be the Bergman space of the disk. In thispaper we characterize the class $\mathcal{A}\subset L^\infty(\mathbb{D})$ such that if $\phi,\psi\in\mathcal{A},\alpha\geq 0$ and $0\leq\phi\leq\alpha\psi$ then there exist positive operators $S,T\in\mathcal{L}(L^2_a(\mathbb{D}))$ such that $\phi(z)=\widetilde{S}(z)\leq\alpha\widetilde{T}(z)=\alpha\psi(z)$ for all $z\in\mathbb{D}$. Further, we have shown that if $S$ and $T$ are two positive operators in $\mathcal{L}(L^2_a(\mathbb{D}))$ and $T$ is invertible then there exists a constant $\alpha\geq0$ such that $\widetilde{S}(z)\leq\alpha\widetilde{T}(z)$ for all $z\in\mathbb{D}$ and $\widetilde{S},\widetilde{T}\in\mathcal{A}$. Here $\mathcal{L}(L^2_a(\mathbb{D}))$ is the space of all boundedlinear operators from $L^2_a(\mathbb{D})$ into $L^2_a(\mathbb{D})$ and $\widetilde{A}(z)=\langle Ak_z,k_z\rangle$ is the Berezintrans form of $A\in\mathcal{L}(L^2_a(\mathbb{D}))$ and $k_z$ is thenormalized reproducing kernel of $L^2_a(\mathbb{D})$. Applications of these results are also obtained.