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

Abstract: The Toeplitz operator acting on the Bergman space $A^{{2}(\mathbb{D})$,} with symbol $\varphi$ is given by $T_{\varphi}f=P(\varphi f)$, 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 $\varphi$ 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.