C[*]-algebras and numerical analysis

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C[*]-algebras and numerical analysis / Roland Hagen, Steffen Roch, Bernd Silbermann

資料種別:

図書

出版情報:

New York : Marcel Dekker, c2001

形態:

376 p. ; 24 cm

シリーズ名:

Monographs and textbooks in pure and applied mathematics ; 236 <BA00046446>

著者名:

ISBN:

9780824704605 [0824704606]

書誌ID:

BA48446749

子書誌情報

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書誌詳細

注記:

Includes bibliographical references (p. 357-371) and index
[*]は上つき文字

言語:

英語

目次情報:

Preface

Introduction / 0：

Numerical analysis / 0.1：

Operator chemistry / 0.2：

The algebraic language of numerical analysis / 0.3：

Microscoping / 0.4：

A few remarks on economy / 0.5：

Brief description of the contents / 0.6：

Approximation methods / 1：

Basic definitions / 1.1.1：

Projection methods / 1.1.2：

Finite section method / 1.1.3：

Banach algebras and stability / 1.2：

Algebras, ideals and homomorphisms / 1.2.1：

Algebraization of stability / 1.2.2：

Small perturbations / 1.2.3：

Compact perturbations / 1.2.4：

Finite sections of Toeplitz operators with continuous generating function / 1.3：

Laurent, Toeplitz and Hankel operators / 1.3.1：

Invertibility and Fredholmness of Toeplitz operators / 1.3.2：

The finite section method / 1.3.3：

C*-algebras of approximation sequences / 1.4：

C*-algebras, their ideals and homomorphisms / 1.4.1：

The Toeplitz C*-algebra and the C*-algebra of the finite section method for Toeplitz operators / 1.4.2：

Stability of sequences in the C*-algebra of the finite section method for Toeplitz operators / 1.4.3：

Symbol of the finite section method for Toeplitz operators / 1.4.4：

Asymptotic behaviour of condition numbers / 1.5：

The condition of an operator / 1.5.1：

Convergence of norms / 1.5.2：

Condition numbers of finite sections of Toeplitz operators / 1.5.3：

Fractality of approximation methods / 1.6：

Fractal homomorphisms, fractal algebras, fractal sequences / 1.6.1：

Fractal algebras, and convergence of norms / 1.6.2：

Notes and references

Regularization of approximation methods / 2：

Stably regularizable sequences / 2.1：

Moore-Penrose inverses and regularizations of matrices / 2.1.1：

Moore-Penrose inverses and regularization of operators / 2.1.2：

Stably regularizable approximation sequences / 2.1.3：

Algebraic characterization of stably regularizable sequences / 2.2：

Moore-Penrose invertibility in C*-algebras / 2.2.1：

Stable regularizability, and Moore-Penrose invertibility in F/G / 2.2.2：

Finite sections of Toeplitz operators and their stable regularizability / 2.2.3：

Convergence of generalized condition numbers / 2.2.4：

Difficulties with Moore-Penrose stability / 2.2.5：

Approximation of spectra / 3：

Set sequences / 3.1：

Limiting sets of set functions / 3.1.1：

Coincidence of the partial and uniform limiting set / 3.1.2：

Spectra and their limiting sets / 3.2：

Limiting sets of spectra of norm convergent sequences / 3.2.1：

Limiting sets of spectra: the general case / 3.2.2：

The case of fractal sequences / 3.2.3：

Limiting sets of singular values / 3.2.4：

Pseudospectra and their limiting sets / 3.3：

[varepsilon]-invertibility / 3.3.1：

Limiting sets of pseudospectra / 3.3.2：

Pseudospectra of operator polynomials / 3.3.3：

Numerical ranges and their limiting sets / 3.4：

Spatial and algebraic numerical ranges / 3.4.1：

Limiting sets of numerical ranges / 3.4.2：

Stability analysis for concrete approximation methods / 3.4.3：

Local principles / 4.1：

Commutative C*-algebras / 4.1.1：

The local principle by Allan and Douglas / 4.1.2：

Fredholmness of Toeplitz operators with piecewise continuous generating function / 4.1.3：

Finite sections of Toeplitz operators generated by a piecewise continuous function / 4.2：

The lifting theorem / 4.2.1：

Application of the local principle / 4.2.2：

Galerkin methods with spline ansatz for singular integral equations / 4.2.3：

Finite sections of Toeplitz operators generated by a quasi-continuous function / 4.3：

Quasicontinuous functions / 4.3.1：

Stability of the finite section method / 4.3.2：

Some other classes of oscillating functions / 4.3.3：

Polynomial collocation methods for singular integral operators with piecewise continuous coefficients / 4.4：

Singular integral operators / 4.4.1：

Stability of the polynomial collocation method / 4.4.2：

Collocation versus Galerkin methods / 4.4.3：

Paired circulants and spline approximation methods / 4.5：

Circulants and paired circulants / 4.5.1：

The stability theorem / 4.5.2：

Finite sections of band-dominated operators / 4.6：

Multidimensional band dominated operators / 4.6.1：

Fredholmness of band dominated operators / 4.6.2：

Finite sections of band dominated operators / 4.6.3：

Representation theory / 5：

Representations / 5.1：

The spectrum of a C*-algebra / 5.1.1：

Primitive ideals / 5.1.2：

The spectrum of an ideal and of a quotient / 5.1.3：

Representations of some concrete algebras / 5.1.4：

Postliminal algebras / 5.2：

Liminal and postliminal algebras / 5.2.1：

Dual algebras / 5.2.2：

Finite sections of Wiener-Hopf operators with almost periodic generating function / 5.2.3：

Lifting theorems and representation theory / 5.3：

Lifting one ideal / 5.3.1：

Sufficient families of homomorphisms / 5.3.2：

Structure of fractal lifting homomorphisms / 5.3.4：

Fredholm sequences / 6：

Fredholm sequences in standard algebras / 6.1：

The standard model / 6.1.1：

Fredholm sequences and stable regularizability / 6.1.2：

Fredholm sequences and Moore-Penrose stability / 6.1.4：

Fredholm sequences and the asymptotic behavior of singular values / 6.2：

The main result / 6.2.1：

A distinguished element and its range dimension / 6.2.2：

Upper estimate of dim Im [Pi subscript n] / 6.2.3：

Lower estimate of dim Im [Pi subscript n] / 6.2.4：

Some examples / 6.2.5：

A general Fredholm theory / 6.3：

Centrally compact and Fredholm sequences / 6.3.1：

Fredholmness modulo compact elements / 6.3.2：

Weakly Fredholm sequences / 6.3.3：

Sequences with finite splitting property / 6.4.1：

Properties of weakly Fredholm sequences / 6.4.2：

Strong limits of weakly Fredholm sequences / 6.4.3：

Weakly Fredholm sequences of matrices / 6.4.4：

Some applications / 6.5：

Numerical determination of the kernel dimension / 6.5.1：

Around the finite section method for Toeplitz operators / 6.5.2：

Discretization of shift operators / 6.5.3：

Self-adjoint approximation sequences / 7：

The spectrum of a self-adjoint approximation sequence / 7.1：

Essential and transient points / 7.1.1：

Fractality of self-adjoint sequences / 7.1.2：

Arveson dichotomy: band operators / 7.1.3：

Arveson dichotomy: standard algebras / 7.1.4：

Szego-type theorems / 7.2：

Folner and Szego algebras / 7.2.1：

Szego's theorem revisited / 7.2.2：

A further generalization of Szego's theorem / 7.2.3：

Algebras with unique tracial state / 7.2.4：

Bibliography

Index

Preface

Introduction / 0：

Numerical analysis / 0.1：

Operator chemistry / 0.2：

The algebraic language of numerical analysis / 0.3：

Microscoping / 0.4：

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