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Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
 
Simon N. Chandler-Wilde University of Reading, Reading, England
Marko Lindner Technical University of Chemnitz, Chemnitz, Germany
Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
eBook ISBN:  978-1-4704-0606-6
Product Code:  MEMO/210/989.E
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
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Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
Simon N. Chandler-Wilde University of Reading, Reading, England
Marko Lindner Technical University of Chemnitz, Chemnitz, Germany
eBook ISBN:  978-1-4704-0606-6
Product Code:  MEMO/210/989.E
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2102011; 111 pp
    MSC: 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 Chandler-Wilde 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2102011; 111 pp
MSC: 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 Chandler-Wilde 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.