Fredholm properties of band-dominated operators on periodic discrete structures
The goal of this paper is to study the essential spectrum of general difference operators (or band operators, with Schrödinger operators as prominent examples) acting on the spaces \(l^p(X)\) on a combinatorial graph \((X,\sim)\). Let \((X,\sim)\) be a combinatorial graph the vertex set \(X\) of which is a discrete metric space. The authors consider that a discrete group \(G\) acts freely on \((X,\sim)\) 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 \(G\). They examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces \(l^p(X)\) or \(c_0(X)\), where \((X,\sim)\) is a periodic graph. The authors' approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [the authors and \textit{B.\,Silbermann}, ``Limit operators and their applications in operator theory'' (Operator Theory:\ Advances and Applications 150; Basel:\ Birkhäuser) (2004; Zbl 1077.47002)] in the special case \(X = G =\mathbb Z^n\) and in [\textit{J.\,Roe}, Integral Equations Oper.\ Theory 51, No.\,3, 411--416 (2005; Zbl 1085.47061)] in the case when \(X = G\) is a general finitely generated discrete group. Sections~4 and 5 are the heart of this paper.
scientific article; zbMATH DE number 5646605