Fredholm properties of band-dominated operators on periodic discrete structures
type: Item
instance of: scholarly article
title: Fredholm properties of band-dominated operators on periodic discrete structures
zbMATH Open document ID: 1191.47039
author: Steffen Roch
author: Vladimir Rabinovich
published in: Complex Analysis and Operator Theory
publication date: 2009-12-14T00:00:00Z
review text: 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.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5646605
zbMATH Keywords: band operators
zbMATH Keywords: discrete graphs
zbMATH Keywords: Fredholmness
zbMATH Keywords: limit operators
zbMATH Keywords: Laplacian
MaRDI profile type: Publication
full work available at URL: s11785-008-0071-0
OpenAlex ID: W2056399867
DOI: 10.1007/S11785-008-0071-0
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description: scientific article; zbMATH DE number 5646605