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

TL;DR

This paper explores the spectral characteristics of Toeplitz operators with harmonic polynomial symbols on the Bergman space, linking integral representations, PDE solutions, and spectral connectedness.

Contribution

It provides new integral representations and conditions for spectra connectedness of Toeplitz operators with harmonic polynomial symbols on the Bergman space.

Findings

Spectra of certain Toeplitz operators are characterized by the image of the symbol.

Fredholm index of these operators can only be m, -m, or 0.

Operators satisfy Coburn's theorem, illustrating properties of non-commuting operators.

Abstract

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.