Hyponormal Toeplitz operators on the weighted Bergman spaces (Q2885348)

Let \(\mathbb D\) be the open unit disk in the complex plane and let \(dA\) denote the normalized area measure on \(\mathbb D\). For \(-1 < \alpha < \infty\), the weighted Bergman space \(A^2_{\alpha}\) is the closed subspace of \(L^2(\mathbb D,dA_{\alpha})\) consisting of all holomorphic functions on \(\mathbb D\), where \(dA_{\alpha}= (\alpha + 1) (1 - |z|^2)^{\alpha} dA(z)\). Let \(P_{\alpha}\) denote the Bergman projection from \(L^2(\mathbb D,dA_{\alpha})\) onto \(A^2_{\alpha}\). For \(u\in L^{\infty}(\mathbb D)\), the Toeplitz operator \(T_u:A^2_{\alpha}\longrightarrow A^2_{\alpha}\) is defined by \(T_uf=P_{\alpha}(uf)\).NEWLINENEWLINEThe authors are interested in studying the hyponormality of Toeplitz operators on \(A^2_{\alpha}.\) A bounded operator \(A\) on a Hilbert space is called hyponormal if \(A^*A \geq AA^*\). The authors study the very special case when the symbol \(u\) is of the form \(u(z) = f(z) + \overline g(z)\), where \(f(z) = a_1z + a_2z^2\) and \(g(z) = b_1z + b_2z^2\). They find a necessary and sufficient condition for \(T_u\) to be hyponormal under the additional assumptions that \(a_1\overline a_2 = b_1\overline b_2\) and \(\alpha \geq 0.\)