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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.23754 |
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| _version_ | 1866910040635473920 |
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| author | Tomar, Nitin |
| author_facet | Tomar, Nitin |
| contents | For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On doubly commuting operators in $C_{1, r}$ class and quantum annulus Tomar, Nitin Functional Analysis 47A15, 47A20, 47A25 For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples. |
| title | On doubly commuting operators in $C_{1, r}$ class and quantum annulus |
| topic | Functional Analysis 47A15, 47A20, 47A25 |
| url | https://arxiv.org/abs/2503.23754 |