Electronic ISBN:  9781470406066 
Product Code:  MEMO/210/989.E 
List Price:  $74.00 
MAA Member Price:  $66.60 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 210; 2011; 111 ppMSC: Primary 47; Secondary 46;
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

Chapters

1. Introduction

2. The Strict Topology

3. Classes of Operators

4. Notions of Operator Convergence

5. Key Concepts and Results

6. Operators on $\ell ^p(\mathbb Z^N,U)$

7. Discrete Schrödinger Operators

8. A Class of Integral Operators

9. Some Open Problems


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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 \(\infty\).

Chapters

1. Introduction

2. The Strict Topology

3. Classes of Operators

4. Notions of Operator Convergence

5. Key Concepts and Results

6. Operators on $\ell ^p(\mathbb Z^N,U)$

7. Discrete Schrödinger Operators

8. A Class of Integral Operators

9. Some Open Problems