International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 472495, 36 pages
http://dx.doi.org/10.1155/2011/472495
Research Article

Multiplication Operators between Lipschitz-Type Spaces on a Tree

Robert F. Allen,1 Flavia Colonna,2 and Glenn R. Easley3

1Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, WI 54601, USA
2Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
3System Planning Corporation, Arlington, VA 22209, USA

Received 4 December 2010; Accepted 16 March 2011

Academic Editor: Ingo Witt

Copyright © 2011 Robert F. Allen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 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 ℒ 𝐰 be the set of functions 𝑓 ∈ 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 ℒ and ℒ 𝐰 and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between ℒ 𝐰 and the space 𝐿 ∞ 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.