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: |