Composition Operators on the Little Lipschitz space of a rooted tree

TL;DR

This paper characterizes bounded composition operators on the little Lipschitz space of a rooted tree, estimates their norms, and investigates their spectral properties and hypercyclicity.

Contribution

It provides a complete characterization of when composition operators are bounded on the space and analyzes their spectral and hypercyclic behavior.

Findings

Characterization of bounded composition operators on the space.

Estimation of operator norms for these operators.

Analysis of the spectrum and hypercyclicity of the operators.

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