Cesàro-like operator acting between Bloch type spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909431756750848 |
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| author | Tang, Pengcheng Zhang, Xuejun |
| author_facet | Tang, Pengcheng Zhang, Xuejun |
| contents | Let $μ$ be a finite positive Borel measure on the interval $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ceàsro-like operator is defined by $$ \mathcal{C}_μ(f)(z)=\sum^\infty_{n=0}μ_n\left(\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $μ_n$ denotes the $n$-th moment of the measure $μ$, that is, $μ_n=\int_{[0, 1)} t^{n}dμ(t)$. In this paper, we characterize the measures $μ$ for which $\mathcal{C}_μ$ is bounded (compact) from one Bloch type space, $\mathcal {B}^α$, into another one, $\mathcal {B}^β$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11921 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cesàro-like operator acting between Bloch type spaces Tang, Pengcheng Zhang, Xuejun Functional Analysis Let $μ$ be a finite positive Borel measure on the interval $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ceàsro-like operator is defined by $$ \mathcal{C}_μ(f)(z)=\sum^\infty_{n=0}μ_n\left(\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $μ_n$ denotes the $n$-th moment of the measure $μ$, that is, $μ_n=\int_{[0, 1)} t^{n}dμ(t)$. In this paper, we characterize the measures $μ$ for which $\mathcal{C}_μ$ is bounded (compact) from one Bloch type space, $\mathcal {B}^α$, into another one, $\mathcal {B}^β$. |
| title | Cesàro-like operator acting between Bloch type spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2208.11921 |