Multiplication Operators between Lipschitz-Type Spaces on a Tree

Abstract EN

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.