Multiplication Operators between Lipschitz-Type Spaces on a Tree

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.