Composition Operators on the Lipschitz Space of a Tree

Abstract

The Lipschitz space \({\mathcal{L}}\) of an infinite tree T rooted at o is defined as the space consisting of the functions \({f : T \rightarrow \mathbb{C}}\) such that $$\beta_f = {\rm sup}\{|f(v) - f(v^-)| : v \in T\backslash\{o\}, \,v^- {\rm parent \, of \,} v\}$$ is finite. Under the norm \({\|f\|_\mathcal{L} = |f(o)|+\beta_f,\mathcal{L}}\) is a Banach space. In this article, the functions φ mapping T into itself whose induced composition operator \({C_{\varphi} : f \mapsto f \circ \varphi}\) on the Lipschitz space is bounded, compact, or an isometry, are characterized. Specifically, it is shown that the symbols of the bounded composition operators are the Lipschitz maps of T into itself viewed as a metric space under the edge-counting distance. The symbols inducing compact operators have finite range while those inducing isometries on \({\mathcal{L}}\) are precisely the onto maps fixing the root and whose images of neighboring vertices coincide or are themselves neighboring vertices. Finally, the spectrum of the operators \({C_\varphi}\) that are isometries is studied in detail.