Multiplication operators between Lipschitz-type spaces on a tree (Q554803)

Summary: Let \(\mathcal L\) 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 \(\mathcal L_w\) 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 paper, we characterize the bounded and the compact multiplication operators between \(\mathcal L\) and \(\mathcal L_w\) and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between \(\mathcal L_w\) 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.