Hyponormal Toeplitz Operators on the Weighted Bergman Space

Abstract.

Consider $\varphi = f + \overline {g}$, where f and g are polynomials, and let $T_{\varphi}$ be the Toeplitz operators with the symbol $\varphi$. It is known that if $T_{\varphi}$ is hyponormal then $|f'(z)|^{2} \geq |g'(z)|^{2}$ 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 $T_{\varphi}$ on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee.