In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space $$\mathbb {T}_{p}$$ T p defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator $$C_\phi $$ C ϕ to be bounded on $${\mathbb {T}}_{p}$$ T p and on $${\mathbb {T}}_{p,0}$$ T p , 0 spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on $${\mathbb {T}}_{p}$$ T p and on $${\mathbb {T}}_{p,0}$$ T p , 0 spaces. Also, we discuss the compactness of $$C_\phi $$ C ϕ on $${\mathbb {T}}_{p}$$ T p and finally prove there are no compact composition operators on $${\mathbb {T}}_{p,0}$$ T p , 0 spaces.

中文翻译：

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同质有根树的 Hardy 空间上的组合算子

在 Muthukumar 和 Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017) 中，本文作者开始研究广义哈代空间 $$\mathbb {T}_{p}$ 离散类似物上的复合算子$ T p 定义在同质有根树上。在这篇文章中，我们给出了组合算子 $$C_\phi $$ C ϕ 在 $${\mathbb {T}}_{p}$$ T p 和 $${\mathbb { 上有界的等价条件T}}_{p,0}$$ T p , 0 个空格并计算它们的算子范数。我们还在 $${\mathbb {T}}_{p}$$ T p 和 $${\mathbb {T}}_{p,0}$$ T p , 0 个空格。此外，我们讨论了 $$C_\phi $$ C ϕ 在 $${\mathbb {T}}_{p}$$ T p 上的紧致性，并最终证明 $${\mathbb { 上没有紧致复合算子T}}_{p,0}$$ T p , 0 个空格。