Multiplication operators on the weighted Lipschitz space of a tree

TL;DR

This paper investigates multiplication operators on a weighted Lipschitz space defined on an infinite tree, providing characterizations of their boundedness, compactness, spectra, and norm estimates, and showing the absence of isometric operators.

Contribution

It offers a comprehensive analysis of multiplication operators on the weighted Lipschitz space over a tree, including new characterizations and spectral properties.

Findings

Characterization of bounded and compact multiplication operators

Norm and essential norm estimates for these operators

Proof that no isometric multiplication operators or zero divisors exist

Abstract

We study the multiplication operators on the weighted Lipschitz space $\mathcal{L}_{\textbf{w}}$ consisting of the complex-valued functions $f$ on the set of vertices of an infinite tree $T$ rooted at $o$ such that $\sup_{v\neq o}|v||f(v)-f(v^-)|<\infty$, where $|v|$ denotes the distance between $o$ and $v$ and $v^-$ is the neighbor of $v$ closest to $o$. For the multiplication operator, we characterize boundedness, compactness, provide estimates on the operator norm and the essential norm, and determine the spectrum. We prove that there are no isometric multiplication operators or isometric zero divisors on $\mathcal{L}_{\textbf{w}}$.