Multiplication Operators between Lipschitz-Type Spaces on a Tree. (26th April 2011) - Allen, Robert F.; Colonna, Flavia; Easley, Glenn R.
Abstract : Letℒ be the space of complex-valued functionsf on the set of verticesT of an infinite tree rooted ato such that the difference of the values off at neighboring vertices remains bounded throughout the tree, and letℒ w be the set of functionsf ∈ ℒ such that| f ( v ) - f ( v - ) | = O ( | v | - 1 ), where| v | is the distance betweeno andv andv - is the neighbor ofv closest too . In this paper, we characterize the bounded and the compact multiplication operators betweenℒ andℒ w and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators betweenℒ w and the spaceL ∞ of bounded functions onT and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.