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9780824704605

C* - Algebras and Numerical Analysis

by ;
  • ISBN13:

    9780824704605

  • ISBN10:

    0824704606

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2000-09-07
  • Publisher: CRC Press

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Summary

"Analyzes algebras of concrete approximation methods detailing prerequisites, local principles, and lifting theorems. Covers fractality and Fredholmness. Explains the phenomena of the asymptotic splitting of the singular values, and more. "

Author Biography

Roland Hagen is a Teacher of Mathematics at Freies Gymnasium Penig, Germany Steffen Roch is a Lecturer at the Technical University of Darmstadt, Germany Bernd Silbermann is Professor of Mathematics, Technical University, Chemnitz, Germany

Table of Contents

Preface 3(8)
Introduction
11(14)
Numerical analysis
11(3)
Operator chemistry
14(1)
The algebraic language of numerical analysis
15(3)
Microscoping
18(3)
A few remarks on economy
21(1)
Brief description of the contents
22(3)
The algebraic language of numerical analysis
25(50)
Approximation methods
25(9)
Basic definitions
26(2)
Projection methods
28(3)
Finite section method
31(3)
Banach algebras and stability
34(11)
Algebras, ideals and homomorphisms
35(1)
Algebraization of stability
36(3)
Small perturbations
39(1)
Compact perturbations
39(6)
Finite sections of Toeplitz operators with continuous generating function
45(7)
Laurent, Toeplitz and Hankel operators
45(3)
Invertibility and Fredholmness of Toeplitz operators
48(1)
The finite section method
49(3)
C*-algebras of approximation sequences
52(10)
C*-algebras, their ideals and homomorphisms
53(3)
The Toeplitz C*-algebra and the C*-algebra of the finite section method for Toeplitz operators
56(4)
Stability of sequences in the C*-algebra of the finite section method for Toeplitz operators
60(1)
Symbol of the finite section method for Toeplitz operators
61(1)
Asymptotic behaviour of condition numbers
62(4)
The condition of an operator
63(1)
Convergence of norms
64(1)
Condition numbers of finite sections of Toeplitz operators
65(1)
Fractality of approximation methods
66(9)
Fractal homomorphisms, fractal algebras, fractal sequences
67(4)
Fractal algebras, and convergence of norms
71(2)
Notes and references
73(2)
Regularization of approximation methods
75(30)
Stably regularizable sequences
76(13)
Moore-Penrose inverses and regularizations of matrices
76(4)
Moore-Penrose inverses and regularization of operators
80(5)
Stably regularizable approximation sequences
85(4)
Algebraic characterization of stably regularizable sequences
89(16)
Moore-Penrose invertibility in C*-algebras
89(3)
Stable regularizability, and Moore-Penrose invertibility in F/G
92(5)
Finite sections of Toeplitz operators and their stable regularizability
97(3)
Convergence of generalized condition numbers
100(3)
Difficulties with Moore-Penrose stability
103(1)
Notes and references
104(1)
Approximation of spectra
105(40)
Set sequences
105(5)
Limiting sets of set functions
106(2)
Coincidence of the partial and uniform limiting set
108(2)
Spectra and their limiting sets
110(9)
Limiting sets of spectra of norm convergent sequences
112(2)
Limiting sets of spectra: the general case
114(3)
The case of fractal sequences
117(2)
Limiting sets of singular values
119(1)
Pseudospectra and their limiting sets
119(15)
ε-invertibility
119(6)
Limiting sets of pseudospectra
125(2)
The case of fractal sequences
127(1)
Pseudospectra of operator polynomials
128(6)
Numerical ranges and their limiting sets
134(11)
Spatial and algebraic numerical ranges
134(2)
Limiting sets of numerical ranges
136(4)
The case of fractal sequences
140(3)
Notes and references
143(2)
Stability analysis for concrete approximation methods
145(62)
Local principles
146(12)
Commutative C*-algebras
146(3)
The local principle by Allan and Douglas
149(2)
Fredholmness of Toeplitz operators with piecewise continuous generating function
151(7)
Finite sections of Toeplitz operators generated by a piecewise continuous function
158(11)
The lifting theorem
158(5)
Application of the local principle
163(4)
Galerkin methods with spline ansatz for singular integral equations
167(2)
Finite sections of Toeplitz operators generated by a quasi-continuous function
169(8)
Quasicontinuous functions
169(4)
Stability of the finite section method
173(2)
Some other classes of oscillating functions
175(2)
Polynomial collocation methods for singular integral operators with piecewise continuous coefficients
177(11)
Singular integral operators
178(5)
Stability of the polynomial collocation method
183(4)
Collocation versus Galerkin methods
187(1)
Paired circulants and spline approximation methods
188(9)
Circulants and paired circulants
190(1)
The stability theorem
191(6)
Finite sections of band-dominated operators
197(10)
Multidimensional band dominated operators
197(1)
Fredholmness of band dominated operators
198(2)
Finite sections of band dominated operators
200(4)
Notes and references
204(3)
Representation theory
207(48)
Representations
208(14)
The spectrum of a C*-algebra
208(2)
Primitive ideals
210(2)
The spectrum of an ideal and of a quotient
212(1)
Representations of some concrete algebras
213(9)
Postliminal algebras
222(16)
Liminal and postliminal algebras
223(3)
Dual algebras
226(4)
Finite sections of Wiener-Hopf operators with almost periodic generating function
230(8)
Lifting theorems and representation theory
238(17)
Lifting one ideal
238(1)
The lifting theorem
239(4)
Sufficient families of homomorphisms
243(6)
Structure of fractal lifting homomorphisms
249(5)
Notes and references
254(1)
Fredholm sequences
255(68)
Fredholm sequences in standard algebras
256(8)
The standard model
256(2)
Fredholm sequences
258(1)
Fredholm sequences and stable regularizability
259(1)
Fredholm sequences and Moore-Penrose stability
260(4)
Fredholm sequences and the asymptotic behavior of singular values
264(18)
The main result
265(1)
A distinguished element and its range dimension
266(3)
Upper estimate of dim Im πn
269(1)
Lower estimate of dim Im πn
270(6)
Some examples
276(6)
A general Fredholm theory
282(23)
Centrally compact and Fredholm sequences
282(6)
Fredholmness modulo compact elements
288(9)
Fredholm sequences in standard algebras
297(8)
Weakly Fredholm sequences
305(9)
Sequences with finite splitting property
305(1)
Properties of weakly Fredholm sequences
305(2)
Strong limits of weakly Fredholm sequences
307(6)
Weakly Fredholm sequences of matrices
313(1)
Some applications
314(9)
Numerical determination of the kernel dimension
314(1)
Around the finite section method for Toeplitz operators
315(1)
Discretization of shift operators
315(7)
Notes and references
322(1)
Self-adjoint approximation sequences
323(30)
The spectrum of a self adjoint approximation sequence
323(16)
Essential and transient points
323(4)
Fractality of self adjoint sequences
327(6)
Arveson dichotomy: band operators
333(5)
Arveson dichotomy: standard algebras
338(1)
Szego-type theorems
339(14)
Folner and Szego algebras
340(6)
Szego's theorem revisited
346(2)
A further generalization of Szego's theorem
348(4)
Algebras with unique tracial state
352(2)
Notes and references
354
Bibliography 353(20)
Index 373

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