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

A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \; \varphi \equiv \bar{f}+g \;\;(f,g\in H^{2})$, 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 note we consider possible versions of this result in the {\it Bergman} space case. \ 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 $m,n,p,q \in \mathbb{Z}_+$, $m < n$ and $p < q$. \ By letting $T_{\varphi}$ act on vectors of the form $$z^k+c z^{\ell}+d z^r \; \; (k<\ell<r),$$ we study the asymptotic behavior of a suitable matrix of inner products, as $k \rightarrow \infty$. \ As a result, we obtain a sharp inequality involving the above mentioned data: $$\left|\alpha \right|^2 n^2 + \left|\beta \right|^2 m^2 - \left|\gamma \right|^2 p^2 - \left|\delta \right|^2 q^2 \ge 2 \left|\bar \alpha \beta m n - \bar \gamma \delta p q \right|.$$ This inequality improves a number of existing results, and it is intended to be a precursor of basic necessary conditions for joint hyponormality of tuples of Toeplitz operators acting on Bergman spaces in one or several complex variables.