Multiplication Operators on the Lipschitz Space of a Tree

Abstract

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space \({\mathcal{L}}\) of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on \({\mathcal{L}}\) that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on \({\mathcal{L}}\) are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of \({\mathcal{L}}\) we call the little Lipschitz space.