Cover image for Infinite matrices and their finite sections : an introduction to the limit operator method
Title:
Infinite matrices and their finite sections : an introduction to the limit operator method
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Publication Information:
Basel : Birkhauser, 2006
ISBN:
9783764377663

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Item Category 1
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PSZ JB 30000010129147 QA188 L56 2006 Open Access Book
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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

Introductionp. vii
1 Preliminariesp. 1
1.1 Basic Conventionsp. 1
1.1.1 Numbers and Vectorsp. 1
1.1.2 Banach Spaces and Banach Algebrasp. 2
1.1.3 Operatorsp. 3
1.2 The Spacesp. 4
1.2.1 Functionsp. 4
1.2.2 Sequencesp. 4
1.2.3 Discretization: Functions as Sequencesp. 5
1.2.4 The System Casep. 5
1.3 The Operatorsp. 5
1.3.1 Operators of Shift and Multiplicationp. 5
1.3.2 Adjoint and Pre-adjoint Operatorsp. 6
1.3.3 An Approximate Identityp. 8
1.3.4 Compact Operators and their Substitutesp. 9
1.3.5 Matrix Representationp. 16
1.3.6 Band- and Band-dominated Operatorsp. 20
1.3.7 Comparisonp. 27
1.4 Invertibility of Sets of Operatorsp. 36
1.5 Approximation Methodsp. 38
1.5.1 Definitionp. 38
1.5.2 Discrete Casep. 39
1.5.3 Continuous Casep. 40
1.5.4 Additional Approximation Methodsp. 40
1.5.5 Which Type of Convergence is Appropriate?p. 41
1.6 {{\cal P}} -convergencep. 41
1.6.1 Definition and Equivalent Characterizationp. 42
1.6.2 {{\cal P}} -convergence in L(E, {{\cal P}} )p. 44
1.6.3 {{\cal P}} -convergence vs. *-strong Convergencep. 46
1.7 Applicability vs. Stabilityp. 47
1.8 Comments and Referencesp. 49
2 Invertibility at Infinityp. 51
2.1 Fredholm Operatorsp. 51
2.2 Invertibility at Infinityp. 53
2.2.1 Invertibility at Infinity in BDO pp. 56
2.3 Invertibility at Infinity vs. Predholmnessp. 56
2.4 Invertibility at Infinity vs. Stabilityp. 59
2.4.1 Stacked Operatorsp. 60
2.4.2 Stability and Stacked Operatorsp. 63
2.5 Comments and Referencesp. 74
3 Limit Operatorsp. 75
3.1 Definition and Basic Propertiesp. 77
3.2 Limit Operators vs. Invertibility at Infinityp. 81
3.2.1 Some Questions Around Theorem 1p. 81
3.2.2 Proof of Theorem 1p. 82
3.3 Fredholmness, Lower Norms and Pseudospectrap. 86
3.3.1 Fredholmness vs. Limit Operatorsp. 86
3.3.2 Pseudospectra vs. Limit Operatorsp. 89
3.4 Limit Operators of a Multiplication Operatorp. 90
3.4.1 Rich Functionsp. 90
3.4.2 Step Functionsp. 93
3.4.3 Bounded and Uniformly Continuous Functionsp. 94
3.4.4 Intermezzo: Essential Cluster Points at Infinityp. 95
3.4.5 Slowly Oscillating Functionsp. 97
3.4.6 Admissible Additive Perturbationsp. 101
3.4.7 Slowly Oscillating and Continuous Functionsp. 101
3.4.8 Almost Periodic Functionsp. 102
3.4.9 Oscillating Functionsp. 106
3.4.10 Pseudo-ergodic Functionsp. 108
3.4.11 Interplay with Convolution Operatorsp. 109
3.4.12 The Big Picturep. 114
3.4.13 Why Choose Integer Sequences?p. 116
3.5 Alternative Views on the Operator Spectrump. 116
3.5.1 The Matrix Point of Viewp. 116
3.5.2 Another Parametrization of the Operator Spectrump. 118
3.6 Generalizations of the Limit Operator Conceptp. 125
3.7 Examplesp. 126
3.7.1 Characteristic Functions of Half Spacesp. 126
3.7.2 Wiener-Hopf and Toeplitz Operatorsp. 128
3.7.3 Singular Integral Operatorsp. 130
3.7.4 Discrete Schrödinger Operatorsp. 132
3.8 Limit Operators - Everything Simple Out There?p. 133
3.8.1 Limit Operators of Limit Operators ofp. 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 Richnessp. 139
3.9.2 Turning Back to the Big Questionp. 140
3.9.3 Alternative Proofs for \ell^{{\infty}}p. 144
3.9.4 Passing to Subclassesp. 145
3.10 Comments and Referencesp. 147
4 Stability of the Finite Section Methodp. 149
4.1 The FSM: Stabihty vs. Limit Operatorsp. 149
4.1.1 Limit Operators of Stacked Operatorsp. 149
4.1.2 The Main Theorem on the Finite Section Methodp. 152
4.1.3 Two Baby Versions of Theorem 4.2p. 154
4.2 The FSM for a Class of Integral Operatorsp. 156
4.2.1 An Algebra of Convolutions and Multiplicationsp. 156
4.2.2 The Finite Section Method in {{\cal A}}_{{\$}}p. 159
4.2.3 A Special Finite Section Method for BCp. 162
4.3 Boundary Integral Equations on Unbounded Surfacesp. 166
4.3.1 The Structure of the Integral Operators Involvedp. 166
4.3.2 Limit Operators of these Integral Operatorsp. 172
4.3.3 A Fredholm Criterion in I + {{\cal K}}_fp. 176
4.3.4 The BC-FSM in I + {{\cal K}}_fp. 176
4.4 Comments and Referencesp. 179
Indexp. 181
Bibliographyp. 185