Positive operators on the Bergman space and Berezin transform

Abstract

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.