Title:

Infinite matrices and their finite sections : an introduction to the limit operator method

Personal Author:

Publication Information:

Basel : Birkhauser, 2006

ISBN:

9783764377663

### Available:*

Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
---|---|---|---|---|---|

PSZ JB | 30000010129147 | QA188 L56 2006 | Open Access Book | Searching... |

### On Order

### Summary

### Summary

This book is concerned with the study of infinite matrices and their approximation by matrices of finite size. The main concepts presented are invertibility at infinity (closely related to Fredholmness), limit operators, and the stability and convergence of finite matrix approximations. Concrete examples are used to illustrate the results throughout, including discrete Schrödinger operators and integral and boundary integral operators arising in mathematical physics and engineering.

### Table of Contents

Introduction | p. vii |

1 Preliminaries | p. 1 |

1.1 Basic Conventions | p. 1 |

1.1.1 Numbers and Vectors | p. 1 |

1.1.2 Banach Spaces and Banach Algebras | p. 2 |

1.1.3 Operators | p. 3 |

1.2 The Spaces | p. 4 |

1.2.1 Functions | p. 4 |

1.2.2 Sequences | p. 4 |

1.2.3 Discretization: Functions as Sequences | p. 5 |

1.2.4 The System Case | p. 5 |

1.3 The Operators | p. 5 |

1.3.1 Operators of Shift and Multiplication | p. 5 |

1.3.2 Adjoint and Pre-adjoint Operators | p. 6 |

1.3.3 An Approximate Identity | p. 8 |

1.3.4 Compact Operators and their Substitutes | p. 9 |

1.3.5 Matrix Representation | p. 16 |

1.3.6 Band- and Band-dominated Operators | p. 20 |

1.3.7 Comparison | p. 27 |

1.4 Invertibility of Sets of Operators | p. 36 |

1.5 Approximation Methods | p. 38 |

1.5.1 Definition | p. 38 |

1.5.2 Discrete Case | p. 39 |

1.5.3 Continuous Case | p. 40 |

1.5.4 Additional Approximation Methods | p. 40 |

1.5.5 Which Type of Convergence is Appropriate? | p. 41 |

1.6 {{\cal P}} -convergence | p. 41 |

1.6.1 Definition and Equivalent Characterization | p. 42 |

1.6.2 {{\cal P}} -convergence in L(E, {{\cal P}} ) | p. 44 |

1.6.3 {{\cal P}} -convergence vs. *-strong Convergence | p. 46 |

1.7 Applicability vs. Stability | p. 47 |

1.8 Comments and References | p. 49 |

2 Invertibility at Infinity | p. 51 |

2.1 Fredholm Operators | p. 51 |

2.2 Invertibility at Infinity | p. 53 |

2.2.1 Invertibility at Infinity in BDO p | p. 56 |

2.3 Invertibility at Infinity vs. Predholmness | p. 56 |

2.4 Invertibility at Infinity vs. Stability | p. 59 |

2.4.1 Stacked Operators | p. 60 |

2.4.2 Stability and Stacked Operators | p. 63 |

2.5 Comments and References | p. 74 |

3 Limit Operators | p. 75 |

3.1 Definition and Basic Properties | p. 77 |

3.2 Limit Operators vs. Invertibility at Infinity | p. 81 |

3.2.1 Some Questions Around Theorem 1 | p. 81 |

3.2.2 Proof of Theorem 1 | p. 82 |

3.3 Fredholmness, Lower Norms and Pseudospectra | p. 86 |

3.3.1 Fredholmness vs. Limit Operators | p. 86 |

3.3.2 Pseudospectra vs. Limit Operators | p. 89 |

3.4 Limit Operators of a Multiplication Operator | p. 90 |

3.4.1 Rich Functions | p. 90 |

3.4.2 Step Functions | p. 93 |

3.4.3 Bounded and Uniformly Continuous Functions | p. 94 |

3.4.4 Intermezzo: Essential Cluster Points at Infinity | p. 95 |

3.4.5 Slowly Oscillating Functions | p. 97 |

3.4.6 Admissible Additive Perturbations | p. 101 |

3.4.7 Slowly Oscillating and Continuous Functions | p. 101 |

3.4.8 Almost Periodic Functions | p. 102 |

3.4.9 Oscillating Functions | p. 106 |

3.4.10 Pseudo-ergodic Functions | p. 108 |

3.4.11 Interplay with Convolution Operators | p. 109 |

3.4.12 The Big Picture | p. 114 |

3.4.13 Why Choose Integer Sequences? | p. 116 |

3.5 Alternative Views on the Operator Spectrum | p. 116 |

3.5.1 The Matrix Point of View | p. 116 |

3.5.2 Another Parametrization of the Operator Spectrum | p. 118 |

3.6 Generalizations of the Limit Operator Concept | p. 125 |

3.7 Examples | p. 126 |

3.7.1 Characteristic Functions of Half Spaces | p. 126 |

3.7.2 Wiener-Hopf and Toeplitz Operators | p. 128 |

3.7.3 Singular Integral Operators | p. 130 |

3.7.4 Discrete Schrödinger Operators | p. 132 |

3.8 Limit Operators - Everything Simple Out There? | p. 133 |

3.8.1 Limit Operators of Limit Operators of | p. 134 |

3.8.2 Everyone is Just a Limit Operator! | p. 136 |

3.9 Big Question: Uniformly or Elementwise? | p. 138 |

3.9.1 Reformulating Richness | p. 139 |

3.9.2 Turning Back to the Big Question | p. 140 |

3.9.3 Alternative Proofs for \ell^{{\infty}} | p. 144 |

3.9.4 Passing to Subclasses | p. 145 |

3.10 Comments and References | p. 147 |

4 Stability of the Finite Section Method | p. 149 |

4.1 The FSM: Stabihty vs. Limit Operators | p. 149 |

4.1.1 Limit Operators of Stacked Operators | p. 149 |

4.1.2 The Main Theorem on the Finite Section Method | p. 152 |

4.1.3 Two Baby Versions of Theorem 4.2 | p. 154 |

4.2 The FSM for a Class of Integral Operators | p. 156 |

4.2.1 An Algebra of Convolutions and Multiplications | p. 156 |

4.2.2 The Finite Section Method in {{\cal A}}_{{\$}} | p. 159 |

4.2.3 A Special Finite Section Method for BC | p. 162 |

4.3 Boundary Integral Equations on Unbounded Surfaces | p. 166 |

4.3.1 The Structure of the Integral Operators Involved | p. 166 |

4.3.2 Limit Operators of these Integral Operators | p. 172 |

4.3.3 A Fredholm Criterion in I + {{\cal K}}_f | p. 176 |

4.3.4 The BC-FSM in I + {{\cal K}}_f | p. 176 |

4.4 Comments and References | p. 179 |

Index | p. 181 |

Bibliography | p. 185 |