Positive Toeplitz operators on the Bergman space (Q1947172)

The paper is devoted to inequalities involving positive Toeplitz operators acting on the Bergman space on the unit disk. Among the results obtained, the authors show that, given a positive Toeplitz operator \(T_{\phi}\) and a finite rank little Hankel operator \(H_{\psi}\) (with \(\psi\) of a special form), there is a bounded linear operator \(A\) such that \(A^*T_{\phi}A \geq S_{\psi}\) and \((\widetilde{A^*T_{\phi}A})(z) \geq \widetilde{S_{\psi}}(z)\), where \(\widetilde{H}\) is the Berezin transform of an operator \(H\); and that, given a non-negative Toeplitz operator \(T_{\phi}\), there exists a rank one operator \(R_1\) such that \(\widetilde{\phi}(z) \geq \beta \widetilde{R_1}(z)\) for some constant \(\beta \geq 0\).