Hyponormal Toeplitz operators with non-harmonic symbol acting on the Bergman space

The Toeplitz operator acting on the Bergman space $A^2(\mathbb{D})$, with symbol $\phi$ is given by $T_{\phi }f=P(\phi f)$, where $P$ is the projection from $L^2(\mathbb{D})$ onto the Bergman space. We present some history on the study of hyponormal Toeplitz operators acting on $A^2(\mathbb{D})$, as well as give results for when $\phi$ is a non-harmonic polynomial. We include a first investigation of Putnam’s inequality for hyponormal operators with non-analytic symbols. Particular attention is given to unusual hyponormality behavior that arises due to the extension of the class of allowed symbols.