A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space

It is a well known result of C. Cowen that, for a symbol $\varphi \in L^{\infty }$, $\varphi =\bar{f}+g\;\;(f,g\in H^{2})$, the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\bar{h}}g$, for some $c\in {\mathbb C}$, $h\in H^{\infty }$, $\left\| h\right\| _{\infty }\leq 1.$ \ In this talk we will consider possible versions of this result in the {it Bergman space} case. \medskip Concretely, we consider Toeplitz operators on the Bergman space of the unit disk, with symbols of the form $$\varphi \equiv \alpha z^n+\beta z^m +\gamma \overline z ^p + \delta \overline z ^q,$$ where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ and $m,n,p,q \in \mathbb{Z}_+$, $m \u3c n$ and $p \u3c q$. \ By letting $T_{\varphi}$ act on vectors of the form $$z^k+c z^{\ell}+d z^r \; \; (k\u3c\el