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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.10771 |
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| _version_ | 1866916483511091200 |
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| author | Augustine, Athul Garayev, M. Shankar, P. |
| author_facet | Augustine, Athul Garayev, M. Shankar, P. |
| contents | For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(Ω)$ over some non-empty set $Ω$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle T\hat{k}_λ,\hat{k}_λ \rangle_{\mathcal{H}} : λ\in Ω\}$ and $\text{ber}(T)$ := $\sup\{|γ|: γ\in \text{Ber}(T)\}$, where $\hat{k}_λ$ is the normalized reproducing kernel for $\mathcal{H}(Ω)$ at $λ\in Ω$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull its Berezin set is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Berezin range and the Berezin radius of some operators Augustine, Athul Garayev, M. Shankar, P. Functional Analysis Operator Algebras For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(Ω)$ over some non-empty set $Ω$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle T\hat{k}_λ,\hat{k}_λ \rangle_{\mathcal{H}} : λ\in Ω\}$ and $\text{ber}(T)$ := $\sup\{|γ|: γ\in \text{Ber}(T)\}$, where $\hat{k}_λ$ is the normalized reproducing kernel for $\mathcal{H}(Ω)$ at $λ\in Ω$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull its Berezin set is discussed. |
| title | On the Berezin range and the Berezin radius of some operators |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2411.10771 |