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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.12551 |
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| _version_ | 1866910749091168256 |
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| author | Taskinen, Jari |
| author_facet | Taskinen, Jari |
| contents | We study spectra of Toeplitz operators $T_a $ with periodic symbols in Bergman spaces $A^2(Π)$ on unbounded periodic planar domains $Π$, which are defined as the union of infinitely many copies of the translated, bounded periodic cell $\varpi$. We introduce Floquet-transform techniques and prove a version of the band-gap-spectrum formula, which is well-known in the framework of periodic elliptic spectral problems and which describes the essential spectrum of $T_a$ in terms of the spectra of a family of Toepliz-type operators $T_{a,η}$ in the cell $\varpi$, where $η$ is the so-called Floquet variable.
As an application, we consider periodic domains $Π_h$ containing thin geometric structures and show how to construct a Toeplitz operator $T_{\sf a}: A^2(Π_h) \to A^2(Π_h)$ such that the essential spectrum of $T_{\sf a}$ contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way how to construct such examples by using Toeplitz operators on the unit disc $\mathbb{D}$ e.g. with radial symbols.
Using a Riemann mapping one can then find a Toeplitz operator $T_a : A^2(\mathbb{D}) \to A^2(\mathbb{D})$ with a bounded symbol and with the same spectral properties as $T_{\sf a}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12551 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Bergman-Toeplitz operators in periodic planar domains Taskinen, Jari Functional Analysis 47A10, 47B35 We study spectra of Toeplitz operators $T_a $ with periodic symbols in Bergman spaces $A^2(Π)$ on unbounded periodic planar domains $Π$, which are defined as the union of infinitely many copies of the translated, bounded periodic cell $\varpi$. We introduce Floquet-transform techniques and prove a version of the band-gap-spectrum formula, which is well-known in the framework of periodic elliptic spectral problems and which describes the essential spectrum of $T_a$ in terms of the spectra of a family of Toepliz-type operators $T_{a,η}$ in the cell $\varpi$, where $η$ is the so-called Floquet variable. As an application, we consider periodic domains $Π_h$ containing thin geometric structures and show how to construct a Toeplitz operator $T_{\sf a}: A^2(Π_h) \to A^2(Π_h)$ such that the essential spectrum of $T_{\sf a}$ contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way how to construct such examples by using Toeplitz operators on the unit disc $\mathbb{D}$ e.g. with radial symbols. Using a Riemann mapping one can then find a Toeplitz operator $T_a : A^2(\mathbb{D}) \to A^2(\mathbb{D})$ with a bounded symbol and with the same spectral properties as $T_{\sf a}$. |
| title | On Bergman-Toeplitz operators in periodic planar domains |
| topic | Functional Analysis 47A10, 47B35 |
| url | https://arxiv.org/abs/2412.12551 |