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

Abstract

A well known result of C. Cowen states that, for a symbol $\phi \in L^{\infty },\phi \equiv \bar{f}+g(f,g\in H^2)$, the Toeplitz operator $T_{\phi }$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\bar{h}}g,$ for some $c\in \mathbb{C}$, $h\in H^{\infty }$, ${\Vert h\Vert }_{\infty }\le 1.$ \ 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 $$\phi \equiv \alpha z^n+\beta z^m+\gamma {\overline{z}}^p+\delta {\overline{z}}^q,$$ where $\alpha ,\beta ,\gamma ,\delta \in \mathbb{C}$ and $m,n,p,q\in {\mathbb{Z}}_{+}$, $m<n$ and $p<q$. \ By letting $T_{\phi }$ 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 $k\to \infty$. \ As a result, we obtain a sharp inequality involving the above mentioned data: $${\left|\alpha \right|}^2{n}^2+{\left|\beta \right|}^2{m}^2-{\left|\gamma \right|}^2{p}^2-{\left|\delta \right|}^2{q}^2\ge 2\left|\bar{\alpha }\beta mn-\bar{\gamma }\delta pq\right|.$$ 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.

16 pages in preprint form; new version includes a minor edit in the statement of Theorem 4.2, and a slightly improved version of the Proof of Thorem 4.2