Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l ^p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case and in [42] in case X = G is a general finitely generated discrete group. Submitted: May 21, 2007. Revised: September 25, 2007. Accepted: November 5, 2007.