Infinite Matrices And Their Finite Sections

This book is concerned with the study of infinite matrices and their approximation by matrices of finite size. Our framework includes the simplest, important case where the matrix entries are numbers, but also the more general case where the entries are bounded linear operators. This generality ensures that examples of the class of operators studied - band-dominated operators on Lebesgue function spaces and sequence spaces - are ubiquitous in mathematics and physics. The main concepts studied 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.This book is written for a broad audience. It will be of particular importance to those researchers and practitioners concerned with large finite matrices and their infinite counterparts, for example in numerical linear algebra and mathematical physics. More generally, the book will be of interest to people working in operator theory and applications, for example studying integral operators or the application of operator algebra methods. While some basic knowledge of functional analysis would be helpful, the presentation contains relevant preliminary material and is largely self-contained.

Introduction	p. vii
Preliminaries	p. 1
Basic Conventions	p. 1
Numbers and Vectors	p. 1
Banach Spaces and Banach Algebras	p. 2
Operators	p. 3
The Spaces	p. 4
Functions	p. 4
Sequences	p. 4
Discretization: Functions as Sequences	p. 5
The System Case	p. 5
The Operators	p. 5
Operators of Shift and Multiplication	p. 5
Adjoint and Pre-adjoint Operators	p. 6
An Approximate Identity	p. 8
Compact Operators and their Substitutes	p. 9
Matrix Representation	p. 16
Band- and Band-dominated Operators	p. 20
Comparison	p. 27
Invertibility of Sets of Operators	p. 36
Approximation Methods	p. 38
Definition	p. 38
Discrete Case	p. 39
Continuous Case	p. 40
Additional Approximation Methods	p. 40
Which Type of Convergence is Appropriate?	p. 41
<$>{\cal P}<$>-convergence	p. 41
Definition and Equivalent Characterization	p. 42
<$>{\cal P}<$>-convergence in L(E,<$>{\cal P}<$>)	p. 44
<$>{\cal P}<$>-convergence vs. *-strong Convergence	p. 46
Applicability vs. Stability	p. 47
Comments and References	p. 49
Invertibility at Infinity	p. 51
Fredholm Operators	p. 51
Invertibility at Infinity	p. 53
Invertibility at Infinity in BDO^p	p. 56
Invertibility at Infinity vs. Predholmness	p. 56
Invertibility at Infinity vs. Stability	p. 59
Stacked Operators	p. 60
Stability and Stacked Operators	p. 63
Comments and References	p. 74
Limit Operators	p. 75
Definition and Basic Properties	p. 77
Limit Operators vs. Invertibility at Infinity	p. 81
Some Questions Around Theorem 1	p. 81
Proof of Theorem 1	p. 82
Fredholmness, Lower Norms and Pseudospectra	p. 86
Fredholmness vs. Limit Operators	p. 86
Pseudospectra vs. Limit Operators	p. 89
Limit Operators of a Multiplication Operator	p. 90
Rich Functions	p. 90
Step Functions	p. 93
Bounded and Uniformly Continuous Functions	p. 94
Intermezzo: Essential Cluster Points at Infinity	p. 95
Slowly Oscillating Functions	p. 97
Admissible Additive Perturbations	p. 101
Slowly Oscillating and Continuous Functions	p. 101
Almost Periodic Functions	p. 102
Oscillating Functions	p. 106
Pseudo-ergodic Functions	p. 108
Interplay with Convolution Operators	p. 109
The Big Picture	p. 114
Why Choose Integer Sequences?	p. 116
Alternative Views on the Operator Spectrum	p. 116
The Matrix Point of View	p. 116
Another Parametrization of the Operator Spectrum	p. 118
Generalizations of the Limit Operator Concept	p. 125
Examples	p. 126
Characteristic Functions of Half Spaces	p. 126
Wiener-Hopf and Toeplitz Operators	p. 128
Singular Integral Operators	p. 130
Discrete Schrödinger Operators	p. 132
Limit Operators - Everything Simple Out There?	p. 133
Limit Operators of Limit Operators of	p. 134
Everyone is Just a Limit Operator!	p. 136
Big Question: Uniformly or Elementwise?	p. 138
Reformulating Richness	p. 139
Turning Back to the Big Question	p. 140
Alternative Proofs for <$>\ell^{\infty}<$>	p. 144
Passing to Subclasses	p. 145
Comments and References	p. 147
Stability of the Finite Section Method	p. 149
The FSM: Stabihty vs. Limit Operators	p. 149
Limit Operators of Stacked Operators	p. 149
The Main Theorem on the Finite Section Method	p. 152
Two Baby Versions of Theorem 4.2	p. 154
The FSM for a Class of Integral Operators	p. 156
An Algebra of Convolutions and Multiplications	p. 156
The Finite Section Method in <$>{\cal A}_{\$}<$>	p. 159
A Special Finite Section Method for BC	p. 162
Boundary Integral Equations on Unbounded Surfaces	p. 166
The Structure of the Integral Operators Involved	p. 166
Limit Operators of these Integral Operators	p. 172
A Fredholm Criterion in <$>I + {\cal K}_f<$>	p. 176
The BC-FSM in <$>I + {\cal K}_f<$>	p. 176
Comments and References	p. 179
Index	p. 181
Bibliography	p. 185
Table of Contents provided by Publisher. All Rights Reserved.

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Infinite Matrices And Their Finite Sections

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Infinite Matrices And Their Finite Sections

Summary

Table of Contents

Supplemental Materials

Digital License