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

TL;DR

This work analyzes the spectral properties of Toeplitz operators on the Bergman space with harmonic symbols, deriving an integral representation for the adjoint $T^*_{z^m}$ that connects Bergman function theory to PDE methods. It develops kernel and range descriptions for operators with symbols of the form $\overline{z}^m+f$, applies Poincaré-difference equation techniques, and uses essential-spectrum theory to establish when spectra are path-connected and how Fredholm indices take the values $m$, $-m$, or $0$. The paper proves that for symbols $\varphi(z)=\overline{z}^m+\alpha z^m+\beta$, the spectrum satisfies $\sigma(T_\varphi)=\overline{\varphi(\mathbb{D})}$, and the index is governed by the zeros of $\alpha t^2+\beta t+1$, yielding Coburn-type behavior in all cases. It introduces the essential projective spectrum framework for non-commuting operator pencils and provides explicit kernel structures, spectrum descriptions, and invertibility criteria across parameter regimes, thereby enriching the understanding of Toeplitz operators on Bergman spaces with harmonic symbols.

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 Poincaré 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 $\varphi (z) =\overline{z}^{m} + αz^m + β$ are characterized, such as $σ(T_\varphi)= \overline{\varphi (\mathbb {D})}$, 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.

Theorems & Definitions (42)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 2.7: Poincaré's Theorem
• Theorem 2.8: Perron's Theorem
• Theorem 2.9: Perron's Theorem