Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space
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| Format: | Preprint |
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2025
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| _version_ | 1866917204774092800 |
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| author | Cui, Puyu Lu, Yufeng Yang, Rongwei Zu, Chao |
| author_facet | Cui, Puyu Lu, Yufeng Yang, Rongwei Zu, Chao |
| contents | 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_φ$ with $φ(z) =\overline{z}^{m} + αz^m + β$ are characterized, such as $σ(T_φ)= \overline{φ(\mathbb {D})}$, Fredholm index of $T_φ$ can only be one of $m,-m$ and $0$, $T_φ$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16952 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space Cui, Puyu Lu, Yufeng Yang, Rongwei Zu, Chao Functional Analysis 47B35, 47B38 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_φ$ with $φ(z) =\overline{z}^{m} + αz^m + β$ are characterized, such as $σ(T_φ)= \overline{φ(\mathbb {D})}$, Fredholm index of $T_φ$ can only be one of $m,-m$ and $0$, $T_φ$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators. |
| title | Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space |
| topic | Functional Analysis 47B35, 47B38 |
| url | https://arxiv.org/abs/2512.16952 |