Introduction to Large Truncated Toeplitz Matrices

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Springer Science & Business Media, 1999 - Mathematics - 258 pages
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Introduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.
 

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Contents

Infinite Matrices
1
12 Laurent Matrices
3
13 Toeplitz Matrices
9
14 Hankel Matrices
13
15 WienerHopf Factorization
15
16 Continuous Symbols
19
17 Locally Sectorial Symbols
20
18 Discontinuous Symbols
25
Determinants and Eigenvalues
121
52 Ising Model and Onsager Formula
127
53 SecondOrder Trace Formulas
132
54 The First Szego Limit Theorem
136
55 Hermitian Toeplitz Matrices
138
56 The AvramParter Theorem
143
57 The Algebraic Approach to Trace Formulas
147
58 Toeplitz Band Matrices
153

Finite Section Method and Stability
31
22 Continuous Symbols
37
23 Asymptotic Inverses
39
24 The GohbergFeldman Approach
44
25 Algebraization of Stability
47
26 Local Principles
52
27 Localization of Stability
56
Norms of Inverses and Pseudospectra
59
32 Continuous Symbols
61
34 Norm of the Resolvent
69
35 Limits of Pseudospectra
70
36 Pseudospectra of Infinite Toeplitz Matrices
80
MoorePenrose Inverses and Singular Values
83
42 The Lowest Singular Value
85
43 The Splitting Phenomenon
86
44 Upper Singular Values
94
45 Molers Phenomenon
95
46 Limiting Sets of Singular Values
98
47 The MoorePenrose Inverse
100
48 Asymptotic MoorePenrose Inversion
108
410 Exact MoorePenrose Sequences
111
411 Regularization and Kato Numbers
116
59 Rational Symbols
163
510 Continuous Symbols
165
511 FisherHartwig Determinants
170
512 Piecewise Continuous Symbols
179
Block Toeplitz Matrices
185
62 Finite Section Method and Stability
191
64 Distribution of Singular Values
197
65 Asymptotic MoorePenrose Inversion
198
66 Trace Formulas
201
67 The SzegoWidom Limit Theorem
203
68 Rational Matrix Symbols
205
69 Multilevel Toeplitz Matrices
208
Banach Space Phenomena
221
72 Fredholmness and Invertibility
222
73 Continuous Symbols
228
74 Piecewise Continuous Symbols
236
75 Loss of Symmetry
240
References
243
Index
255
Symbol Index
257
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About the author (1999)

Bottcher of the Technical University of Chemnitz, Germany

Silbermann of the Technical University of Chemintz, Germany

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