C*-algebras and numerical analysis /

This book is about C*-algebras as a tool in numerical analysis.

Bibliographic Details
Main Author:	Hagen, Roland, 1953-
Corporate Author:	Taylor & Francis
Other Authors:	Roch, Steffen, 1958-, Silbermann, Bernd, 1941-
Format:	eBook
Language:	English
Published:	New York : Marcel Dekker, ©2001.
Series:	Monographs and textbooks in pure and applied mathematics ; 236.
Subjects:	Numerical analysis. C-algebras. Analyse numérique. C-algèbres. MATHEMATICS > Algebra > Linear. C*-algebras Numerical analysis Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

0.1 Numerical analysis 11
0.2 Operator chemistry 14
0.3 Algebraic language of numerical analysis 15
0.4 Microscoping 18
0.5 A few remarks on economy 21
0.6 Brief description of the contents 22
1 Algebraic language of numerical analysis 25
1.1 Approximation methods 25
1.1.1 Basic definitions 26
1.1.2 Projection methods 28
1.1.3 Finite section method 31
1.2 Banach algebras and stability 34
1.2.1 Algebras, ideals and homomorphisms 35
1.2.2 Algebraization of stability 36
1.2.3 Small perturbations 39
1.2.4 Compact perturbations 39
1.3 Finite sections of Toeplitz operators with continuous generating function 45
1.3.1 Laurent, Toeplitz and Hankel operators 45
1.3.2 Invertibility and Fredholmness of Toeplitz operators 48
1.3.3 Finite section method 49
1.4 C*-algebras of approximation sequences 52
1.4.1 C*-algebras, their ideals and homomorphisms 53
1.4.2 Toeplitz C*-algebra and the C*-algebra of the finite section method for Toeplitz operators 56
1.4.3 Stability of sequences in the C*-algebra of the finite section method for Toeplitz operators 60
1.4.4 Symbol of the finite section method for Toeplitz operators 61
1.5 Asymptotic behaviour of condition numbers 62
1.5.1 Condition of an operator 63
1.5.2 Convergence of norms 64
1.5.3 Condition numbers of finite sections of Toeplitz operators 65
1.6 Fractality of approximation methods 66
1.6.1 Fractal homomorphisms, fractal algebras, fractal sequences 67
1.6.2 Fractal algebras, and convergence of norms 71
2 Regularization of approximation methods 75
2.1 Stably regularizable sequences 76
2.1.1 Moore-Penrose inverses and regularizations of matrices 76
2.1.2 Moore-Penrose inverses and regularization of operators 80
2.1.3 Stably regularizable approximation sequences 85
2.2 Algebraic characterization of stably regularizable sequences 89
2.2.1 Moore-Penrose invertibility in C*-algebras 89
2.2.2 Stable regularizability, and Moore-Penrose invertibility in F/G 92
2.2.3 Finite sections of Toeplitz operators and their stable regularizability 97
2.2.4 Convergence of generalized condition numbers 100
2.2.5 Difficulties with Moore-Penrose stability 103
3 Approximation of spectra 105
3.1 Set sequences 105
3.1.1 Limiting sets of set functions 106
3.1.2 Coincidence of the partial and uniform limiting set 108
3.2 Spectra and their limiting sets 110
3.2.1 Limiting sets of spectra of norm convergent sequences 112
3.2.2 Limiting sets of spectra: the general case 114
3.2.3 Case of fractal sequences 117
3.2.4 Limiting sets of singular values 119
3.3 Pseudospectra and their limiting sets 119
3.3.1 [varepsilon]-invertibility 119
3.3.2 Limiting sets of pseudospectra 125
3.3.3 Case of fractal sequences 127
3.3.4 Pseudospectra of operator polynomials 128
3.4 Numerical ranges and their limiting sets 134
3.4.1 Spatial and algebraic numerical ranges 134
3.4.2 Limiting sets of numerical ranges 136
3.4.3 Case of fractal sequences 140
4 Stability analysis for concrete approximation methods 145
4.1 Local principles 146
4.1.1 Commutative C*-algebras 146
4.1.2 Local principle by Allan and Douglas 149
4.1.3 Fredholmness of Toeplitz operators with piecewise continuous generating function 151
4.2 Finite sections of Toeplitz operators generated by a piecewise continuous function 158
4.2.1 Lifting theorem 158
4.2.2 Application of the local principle 163
4.2.3 Galerkin methods with spline ansatz for singular integral equations 167
4.3 Finite sections of Toeplitz operators generated by a quasi-continuous function 169
4.3.1 Quasicontinuous functions 169
4.3.2 Stability of the finite section method 173
4.3.3 Some other classes of oscillating functions 175
4.4 Polynomial collocation methods for singular integral operators with piecewise continuous coefficients 177
4.4.1 Singular integral operators 178
4.4.2 Stability of the polynomial collocation method 183
4.4.3 Collocation versus Galerkin methods 187
4.5 Paired circulants and spline approximation methods 188
4.5.1 Circulants and paired circulants 190
4.5.2 Stability theorem 191
4.6 Finite sections of band-dominated operators 197
4.6.1 Multidimensional band dominated operators 197
4.6.2 Fredholmness of band dominated operators 198
4.6.3 Finite sections of band dominated operators 200
5 Representation theory 207
5.1 Representations 208
5.1.1 Spectrum of a C*-algebra 208
5.1.2 Primitive ideals 210
5.1.3 Spectrum of an ideal and of a quotient 212
5.1.4 Representations of some concrete algebras 213
5.2 Postliminal algebras 222
5.2.1 Liminal and postliminal algebras 223
5.2.2 Dual algebras 226
5.2.3 Finite sections of Wiener-Hopf operators with almost periodic generating function 230
5.3 Lifting theorems and representation theory 238
5.3.1 Lifting one ideal 238
5.3.2 Lifting theorem 239
5.3.3 Sufficient families of homomorphisms 243
5.3.4 Structure of fractal lifting homomorphisms 249
6 Fredholm sequences 255
6.1 Fredholm sequences in standard algebras 256
6.1.1 Standard model 256
6.1.2 Fredholm sequences 258
6.1.3 Fredholm sequences and stable regularizability 259
6.1.4 Fredholm sequences and Moore-Penrose stability 260
6.2 Fredholm sequences and the asymptotic behavior of singular values 264
6.2.1 Main result 265
6.2.2 A distinguished element and its range dimension 266
6.2.3 Upper estimate of dim Im [Pi subscript n] 269
6.2.4 Lower estimate of dim Im [Pi subscript n] 270
6.2.5 Some examples 276
6.3 A general Fredholm theory 282
6.3.1 Centrally compact and Fredholm sequences 282
6.3.2 Fredholmness modulo compact elements 288
6.3.3 Fredholm sequences in standard algebras 297
6.4 Weakly Fredholm sequences 305
6.4.1 Sequences with finite splitting property 305
6.4.2 Properties of weakly Fredholm sequences 305
6.4.3 Strong limits of weakly Fredholm sequences 307
6.4.4 Weakly Fredholm sequences of matrices 313
6.5 Some applications 314
6.5.1 Numerical determination of the kernel dimension 314
6.5.2 Around the finite section method for Toeplitz operators 315
6.5.3 Discretization of shift operators 317
7 Self-adjoint approximation sequences 323
7.1 Spectrum of a self-adjoint approximation sequence 323
7.1.1 Essential and transient points 323
7.1.2 Fractality of self-adjoint sequences 327
7.1.3 Arveson dichotomy: band operators 333
7.1.4 Arveson dichotomy: standard algebras 338
7.2 Szego-type theorems 339
7.2.1 Folner and Szego algebras 340
7.2.2 Szego's theorem revisited 346
7.2.3 A further generalization of Szego's theorem 348
7.2.4 Algebras with unique tracial state 352.