Hyponormal Toeplitz Operators on the Weighted Bergman Space

Consider j = f +[`(g)]varphi = f + overline {g}, where f and g are polynomials, and let T_jT_{varphi} be the Toeplitz operators with the symbol jvarphi. It is known that if T_jT_{varphi} is hyponormal then |f￠(z)|² 3 |g￠(z)|²|f'(z)|^{2} geq |g'(z)|^{2} on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of T_jT_{varphi} on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee.