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Composition operators on Hardy spaces of the homogenous rooted trees. (English) Zbl 1508.47048

Summary: The authors [Bull. Malays. Math. Sci. Soc. (2) 40, No. 4, 1801–1815 (2017; Zbl 1508.47047)] initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb{T}_p\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi\) to be bounded on \(\mathbb{T}_p\) and on \(\mathbb{T}_{p, 0}\) spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on \(\mathbb{T}_p\) and on \(\mathbb{T}_{p, 0}\) spaces. Also, we discuss the compactness of \(C_\phi\) on \(\mathbb{T}_p\) and finally prove there are no compact composition operators on \(\mathbb{T}_{p, 0}\) spaces.

MSC:

47B33 Linear composition operators
05C05 Trees
39A12 Discrete version of topics in analysis
30H10 Hardy spaces
46B50 Compactness in Banach (or normed) spaces

Citations:

Zbl 1508.47047
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References:

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