Composition Operators on the Lipschitz Space of a Tree

The Lipschitz space of an infinite tree T rooted at o is defined as the space consisting of the functions f : T -> C such that beta(f) = sup{vertical bar f(v) - f(v(-))vertical bar : v is an element of T\{0}, v(-) parent of v} is finite. Under the norm parallel to f parallel to() pound = vertical bar f(0)vertical bar + beta(f), pound is a Banach space. In this article, the functions phi mapping T into itself whose induced composition operator 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 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 that are isometries is studied in detail.