Multiplication operators between Lipschitz-type spaces on a tree

TL;DR

This paper characterizes bounded and compact multiplication operators between Lipschitz-type spaces on an infinite rooted tree, providing norm estimates and showing the absence of isometries among these operators.

Contribution

It offers a complete characterization of multiplication operators between specific Lipschitz-type spaces on trees, including norm and compactness criteria, which was previously unexplored.

Findings

Characterized bounded and compact multiplication operators between al and w spaces.

Provided operator norm and essential norm estimates for these operators.

Proved there are no isometries among the multiplication operators studied.

Abstract

Let $\mathcal{L}$ be the space of complex-valued functions $f$ on the set of vertices $T$ of an rooted infinite tree rooted at $o$ such that the difference of the values of $f$ at neighboring vertices remains bounded throughout the tree, and let $\mathcal{L}_{\textbf{w}}$ be the set of functions $f\in \mathcal{L}$ such that $|f(v)-f(v^-)|=O(|v|^{-1})$, where $|v|$ is the distance between $o$ and $v$ and $v^-$ is the neighbor of $v$ closest to $o$. In this article, we characterize the bounded and the compact multiplication operators between $\mathcal{L}$ and $\mathcal{L}_{\textbf{w}}$, and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between $\mathcal{L}_{\textbf{w}}$ and the space $L^\infty$ of bounded functions on $T$ and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.