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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2401.03176 |
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| _version_ | 1866913187757031424 |
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| author | Augustine, Athul Garayev, M. Shankar, P. |
| author_facet | Augustine, Athul Garayev, M. Shankar, P. |
| contents | The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convexity of the Berezin range of composition operators and related questions Augustine, Athul Garayev, M. Shankar, P. Functional Analysis Complex Variables 47B32, 52A10 The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$. |
| title | On the convexity of the Berezin range of composition operators and related questions |
| topic | Functional Analysis Complex Variables 47B32, 52A10 |
| url | https://arxiv.org/abs/2401.03176 |