Cesàro-type operators acting on Dirichlet spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866913971071614976 |
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| author | Blasco, Óscar Galanopoulos, Petros Girela, Daniel |
| author_facet | Blasco, Óscar Galanopoulos, Petros Girela, Daniel |
| contents | If $(η)=\{ η_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Cesàro-type operator $\mathcal C_{(η)}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb D$ as follows: If $f$ is an analytic function in $\mathbb D$, $f(z)=\sum_{n=0}^\infty a_nz^n$ ($z\in \mathbb D$), then $\mathcal C_{(η)}(f)$ is formally defined by $$\mathcal C_{(η)}(f)(z)=\mathcal C_{\{η_n\}}(f)(z)=\sum_{n=0}^\infty η_n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(η)}$ is a natural generalization of the Cesàro operator. For each $α\in \mathbb R$ we let $\mathcal D^2_α$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-α} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. In this paper we give a complete characterization of the sequences of complex numbers $(η)$ for which the operator $\mathcal C_{(η)}$ is bounded (compact) from $\mathcal D^2_α$ into $\mathcal D^2_β$ for any $α, β\in \mathbb R$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cesàro-type operators acting on Dirichlet spaces Blasco, Óscar Galanopoulos, Petros Girela, Daniel Complex Variables 47B91, 30H20, 30H25 If $(η)=\{ η_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Cesàro-type operator $\mathcal C_{(η)}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb D$ as follows: If $f$ is an analytic function in $\mathbb D$, $f(z)=\sum_{n=0}^\infty a_nz^n$ ($z\in \mathbb D$), then $\mathcal C_{(η)}(f)$ is formally defined by $$\mathcal C_{(η)}(f)(z)=\mathcal C_{\{η_n\}}(f)(z)=\sum_{n=0}^\infty η_n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(η)}$ is a natural generalization of the Cesàro operator. For each $α\in \mathbb R$ we let $\mathcal D^2_α$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-α} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. In this paper we give a complete characterization of the sequences of complex numbers $(η)$ for which the operator $\mathcal C_{(η)}$ is bounded (compact) from $\mathcal D^2_α$ into $\mathcal D^2_β$ for any $α, β\in \mathbb R$. |
| title | Cesàro-type operators acting on Dirichlet spaces |
| topic | Complex Variables 47B91, 30H20, 30H25 |
| url | https://arxiv.org/abs/2507.13502 |