Composition Operators on the Little Lipschitz space of a rooted tree

Abstract

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 $$\underset{|v|\to \infty }{\lim }|f(v)-f(v^{-})|=0,$$ 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 $\varphi$ of $T$ for which the composition operator $C_{\varphi }$ is bounded and we estimate its operator norm. In addition, we study the spectrum of $C_{\varphi }$ and the hypercyclicity of the operators $\lambda C_{\varphi }$ for $\lambda \in \mathbb{C}$.