Positive operators on the Bergman space and Berezin transform (Q2896638)

Let \(\mathbb D\) be the open unit disk in \(\mathbb C\), \(L_a^2(\mathbb D)\) be the Bergman space of analytic functions on \(\mathbb D\). The Berezin transform of a bounded linear operator \(A\) on \(L_a^2(\mathbb D)\) is the function \(\tilde{A}(z)=\langle Ak_z,k_z\rangle\), \(z\in \mathbb Z\), where \(k_z\) is the normalized reproducing kernel.NEWLINENEWLINEThe authors characterize the class \(\mathcal A\subset L^\infty (\mathbb D)\) of such functions that if \(\varphi ,\psi \in \mathcal A\), \(\alpha \geq 0\), and \(0\leq \varphi \leq \alpha \psi\), then there exist such positive operators \(S,T\) on \(L_a^2(\mathbb D)\) that \(\varphi (z)=\tilde{S}(z)\leq \alpha \tilde{T}(z)=\alpha \psi (z)\) for all \(z\in \mathbb D\). In addition, for any positive operators \(S\) and \(T\) such that \(T\) is invertible, it is shown that \(\tilde{S}(z)\leq \alpha \tilde{T}(z)\) for all \(z\in \mathbb D\) and some constant \(\alpha \geq 0\). The case of hyponormal operators on \(L_a^2(\mathbb D)\) is considered in detail.