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Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices

Marko Lindner Technical University of Chemnitz, Chemnitz, Germany
Available Formats:
Electronic 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 Click above image for expanded view 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 Available Formats:  Electronic 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 $\infty$.

• 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

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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 reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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