Generalized complex symmetric composition operators with applications
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Allu, Vasudevarao Sahoo, Satyajit |
| author_facet | Allu, Vasudevarao Sahoo, Satyajit |
| contents | We characterize the weighted composition-differentiation operators $D_{\mfn,ψ,φ}$ acting on $\mathcal{H}_γ(\mathbb{D}^d)$ over the polydisk $\mathbb{D}^d$ which are complex symmetric with respect to the conjugation $\mathcal{J}$. We obtain necessary and sufficient conditions for $D_{\mfn,ψ,φ}$ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators $M_{n, ψ, φ}=\displaystyle\sum_{j=1}^{n}a_jD_{j,ψ_j, φ},$ (where $a_j\in \mathbb{C}$ for $j=1, 2, \dots, n$) on the reproducing kernel Hilbert space $\mathcal{H}_γ(\mathbb{D})$ of analytic functions on the unit disk $\mathbb{D}$ with respect to a weighted composition conjugation $C_{μ, ξ}$. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on $\mathcal{H}_γ(\mathbb{D})$ are investigated. Additionally, geometrical interpretations have also been employed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized complex symmetric composition operators with applications Allu, Vasudevarao Sahoo, Satyajit Functional Analysis 47B33, 47B15, 47A12, 47B32 We characterize the weighted composition-differentiation operators $D_{\mfn,ψ,φ}$ acting on $\mathcal{H}_γ(\mathbb{D}^d)$ over the polydisk $\mathbb{D}^d$ which are complex symmetric with respect to the conjugation $\mathcal{J}$. We obtain necessary and sufficient conditions for $D_{\mfn,ψ,φ}$ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators $M_{n, ψ, φ}=\displaystyle\sum_{j=1}^{n}a_jD_{j,ψ_j, φ},$ (where $a_j\in \mathbb{C}$ for $j=1, 2, \dots, n$) on the reproducing kernel Hilbert space $\mathcal{H}_γ(\mathbb{D})$ of analytic functions on the unit disk $\mathbb{D}$ with respect to a weighted composition conjugation $C_{μ, ξ}$. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on $\mathcal{H}_γ(\mathbb{D})$ are investigated. Additionally, geometrical interpretations have also been employed. |
| title | Generalized complex symmetric composition operators with applications |
| topic | Functional Analysis 47B33, 47B15, 47A12, 47B32 |
| url | https://arxiv.org/abs/2502.20875 |