Hyponormal Toeplitz operators on the weighted Bergman space
A bounded linear operator \(A\) on a Hilbert space is called hyponormal if \(A^*A-AA^*\) is a positive operator. The authors of the present paper study hyponormality of Toeplitz operators on the weighted Bergman space of the unit disk \(\mathbb{D}\). If \(dA\) denotes the normalized Lebesgue area measure on \(\mathbb{D}\) and if \(-1<\alpha<\infty\), let \(dA_{\alpha}(z)=(\alpha+1)(1-|z|^2)^{\alpha}\,dA(z)\). The weighted Bergman space \(A^2_{\alpha}\) is the closed subspace of \(L^2(\mathbb{D},dA_{\alpha})\) consisting of analytic functions on \(\mathbb{D}\). Let \(P_{\alpha}\) denote the orthogonal projection from \(L^2(\mathbb{D},dA_{\alpha})\) onto \(A^2_{\alpha}\). For \(\varphi\in L^{\infty}(\mathbb{D})\), the Toeplitz operator \(T_{\varphi}\) is defined on \(A^2_{\alpha}\) by \(T_{\varphi}h=P_{\alpha}(\varphi h)\). \textit{C.\,Cowen} [``Hyponormality of Toeplitz operators'', Proc.\ Am.\ Math.\ Soc.\ 103, No.\,3, 809--812 (1988; Zbl 0668.47021)] gave a characterization of hyponormality of Toeplitz operators on the Hardy space. \textit{P.\,Ahern} and \textit{Ž.\,Čučković} [``A mean value inequality with applications to Bergman space operators'', Pac.\ J.\ Math.\ 173, No.\,2, 295--305 (1996; Zbl 0962.47015)] showed that, if \(\varphi =f+\bar{g}\) is bounded, with \(f\) and \(g\) analytic on \(\mathbb{D}\), then the hyponormality of \(T_{\varphi}\) on \(A^2_{\alpha}\) implies that \(\limsup_{z\rightarrow\xi}(|f'(z)|^2-|g'(z)|^2)\geq 0\) for all \(\xi\in\partial\mathbb{D}\). The authors of the present paper show that this condition is not sufficient. They find a necessary condition for hyponormality of \(T_{\varphi}\) in the case when \(f\) and \(g\) are polynomials. In a special case, they find a necessary and sufficient condition for the hyponormality of \(T_{\varphi}\) on \(A^2_{\alpha}\).
scientific article; zbMATH DE number 5617587