Positive Toeplitz operators on the Bergman space

Abstract

‎In this paper we find conditions on the existence of bounded linear‎ ‎operators $A$ on the Bergman space $L_{a}^{2}(\mathbb{D})$ such that‎ ‎$A^{\ast}T_{\phi}A\geq S_{\psi}$ and $A^{\ast}T_{\phi}A\geq‎ ‎T_{\phi}$ where $T_{\phi}$ is a positive Toeplitz operator on‎ ‎$L_{a}^{2}(\mathbb{D})$ and $S_{\psi}$ is a self-adjoint little‎ ‎Hankel operator on $L_{a}^{2}(\mathbb{D})$ with symbols $\phi‎, ‎\psi\in L^{\infty}(\mathbb{D})$ respectively‎. ‎Also we show that if‎ ‎$T_{\phi}$ is a non-negative Toeplitz operator then there exists a‎ ‎rank one operator $R_{1}$ on $L_{a}^{2}(\mathbb{D})$ such that‎ ‎$\widetilde{\phi}(z)\geq \alpha^{2}\widetilde{R_{1}}(z)$ for some‎ ‎constant $\alpha\geq 0$ and for all $z\in \mathbb{D}$ where‎ ‎$\widetilde{\phi}$ is the Berezin transform of $T_{\phi}$ and‎ ‎$\widetilde{R_{1}}(z)$ is the Berezin transform of $R_{1}$‎.