Multiplication Operators between Lipschitz-Type Spaces on a Tree

Abstract

Let ℒ be the space of complex-valued functions 𝑓 on the set of vertices 𝑇 of an infinite tree rooted at 𝑜 such that the difference of the values of 𝑓 at neighboring vertices remains bounded throughout the tree, and let ℒ𝐰 be the set of functions 𝑓∈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 paper, we characterize the bounded and the compact multiplication operators between ℒ and ℒ𝐰 and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between ℒ𝐰 and the space 𝐿∞ 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.