Composition operators on Hardy spaces of the homogenous rooted trees

Abstract

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 \(\mathbb {T}_{p}\) 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.