Multiplication Operators between Lipschitz-Type Spaces on a Tree

Let â„’ be the space of complex-valued functions ð ‘“ on the set of vertices 𠑇 of an infinite tree rooted at ð ‘œ such that the difference of the values of ð ‘“ at neighboring vertices remains bounded throughout the tree, and let â„’ ð ° be the set of functions ð ‘“ ∈ â„’ such that | ð ‘“ ( ð ‘£ ) − ð ‘“ ( ð ‘£ − ) | = ð ‘‚ ( | ð ‘£ | − 1 ) , where | ð ‘£ | is the distance between ð ‘œ and ð ‘£ and ð ‘£ − is the neighbor of ð ‘£ closest to ð ‘œ . In this paper, we characterize the bounded and the compact multiplication operators between â„’ and â„’ ð ° and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between â„’ ð ° and the space ð ¿ âˆž of bounded functions on 𠑇 and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.