Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space

This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $T^*_{z^m}$, which establishes a connection between the Bergman functions and the solutions of PDE theory. In fact, by leveraging the Poincar\'e theorem in difference equations and the solution forms of differential equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols, and further gives the sufficient conditions for the connectedness of the spectra of these Toeplitz operators. The spectral properties of $T_\varphi$ with $\varphi (z) =\overline{z}^{m} + \alpha z^m + \beta$ are characterized, such as $\sigma(T_\varphi)= \overline{\varphi (\mathbb {D})}$, Fredholm index of $T_\varphi$ can only be one of $m,-m$ and $0$, $T_\varphi$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators.