Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols

Abstract

Let $p$ be an analytic polynomial on the unit disk. We obtain a necessary and sufficient condition for Toeplitz operators with the symbol $\overline{z}+p$ to be invertible on the Bergman space when all coefficients of $p$ are real numbers. Furthermore, we establish several necessary and sufficient, easy-to-check conditions for Toeplitz operators with the symbol $\overline{z}+p$ to be invertible on the Bergman space when some coefficients of $p$ are complex numbers.