Multiplication Operators between Lipschitz-Type Spaces on a Tree. (5th June 2011) - Allen, Robert F.; Colonna, Flavia; Easley, Glenn R.
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.