Composition Operators on the Lipschitz Space of a Tree |
| |
Authors: | Robert F Allen Flavia Colonna Glenn R Easley |
| |
Institution: | 1. Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, WI, 54601, USA 2. Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA, 22030, USA 3. MITRE, 7515 Colshire Drive, McLean, VA, 22102, USA
|
| |
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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|