Memoirs of the American Mathematical Society 2011; 111 pp; softcover Volume: 210 ISBN10: 0821852434 ISBN13: 9780821852439 List Price: US$74 Individual Members: US$44.40 Institutional Members: US$59.20 Order Code: MEMO/210/989
 In the first half of this memoir the authors explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by ChandlerWilde and Zhang (2002). They build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator \(A\) (its operator spectrum). In the second half of this memoir the authors study bounded linear operators on the generalised sequence space \(\ell^p(\mathbb{Z}^N,U)\), where \(p\in [1,\infty]\) and \(U\) is some complex Banach space. They make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator \(A\) is a locally compact perturbation of the identity. Especially, they obtain stronger results than previously known for the subtle limiting cases of \(p=1\) and \(\infty\). Table of Contents  Introduction
 The strict topology
 Classes of operators
 Notions of operator convergence
 Key concepts and results
 Operators on \(\ell^p(\mathbb Z^N,U)\)
 Discrete Schrödinger operators
 A class of integral operators
 Some open problems
 Bibliography
 Index
