Composition Operators on the Little Lipschitz space of a rooted tree
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Colonna, Flavia Martínez-Avendaño, Rubén A. |
| author_facet | Colonna, Flavia Martínez-Avendaño, Rubén A. |
| contents | In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$, defined as the subspace of the Lipschitz space ${\mathcal L}$ consisting of the complex-valued functions $f$ on $T$ such that $$\lim_{|v|\to\infty}|f(v)-f(v^-)|=0,$$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give a complete characterization of the self-maps $ϕ$ of $T$ for which the composition operator $C_ϕ$ is bounded and we estimate its operator norm. In addition, we study the spectrum of $C_ϕ$ and the hypercyclicity of the operators $λC_ϕ$ for $λ\in {\mathbb C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14714 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Composition Operators on the Little Lipschitz space of a rooted tree Colonna, Flavia Martínez-Avendaño, Rubén A. Functional Analysis 47A16, 47B37, 05C05, 05C63 In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$, defined as the subspace of the Lipschitz space ${\mathcal L}$ consisting of the complex-valued functions $f$ on $T$ such that $$\lim_{|v|\to\infty}|f(v)-f(v^-)|=0,$$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give a complete characterization of the self-maps $ϕ$ of $T$ for which the composition operator $C_ϕ$ is bounded and we estimate its operator norm. In addition, we study the spectrum of $C_ϕ$ and the hypercyclicity of the operators $λC_ϕ$ for $λ\in {\mathbb C}$. |
| title | Composition Operators on the Little Lipschitz space of a rooted tree |
| topic | Functional Analysis 47A16, 47B37, 05C05, 05C63 |
| url | https://arxiv.org/abs/2410.14714 |