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Lindner M. — Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method
Lindner M. — Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method

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Название: Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method

Автор: Lindner M.


This book is an introduction to a fascinating topic at the interface of functional analysis, algebra and numerical analysis, written for a broad audience of students, researchers and practitioners. It is concerned with the study of infinite matrices and their approximation by matrices of finite size. Our framework includes the simplest, important case where the matrix entries are numbers, but also the more general case where the entries are bounded linear operators. This ensures that examples of the class of operators studied - band-dominated operators on Lebesgue function spaces and sequence spaces - are ubiquitous in mathematics and physics.

The main items and concepts studied are band-dominated operators, invertibility at infinity, Fredholmness, the method of limit operators, and the stability and convergence of finite matrix approximations. Concrete examples are used to illustrate the results throughout, including discrete Schrödinger operators and integral and boundary integral operators arising in mathematical physics and engineering.

The main audience for this book are people concerned with large finite matrices and their infinite counterparts, for example in numerical linear algebra and mathematical physics. More generally, the book will be of interest to those working in operator theory and applications, for example studying integral operators or the application of operator algebra methods. While some basic knowledge of functional analysis would be helpful, the presentation contains relevant preliminary material and is largely self-contained.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 191

Добавлена в каталог: 02.07.2008

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Предметный указатель
*-strong convergence      10
Adjoint operator      7
Almost periodic      103 104
Applicability      39
Approximate identity      9
Approximation method      38
Banach algebra      2
Band matrix      20
Band operator      20
Band width      20
Band-dominated operator      21
BC-FSM      164
Bounded below      69
Center      122
Central localization      122
CHARACTER      118
Coefficients of a band operator      21
Coefficients of a band-dom op      21
column      19
Commutator      3
Convergence, $\mathcal{P}-$      42
Convergence, *-strong      10
Convergence, K-strong      10
Convergence, norm-      3
Convergence, pre*-strong      10
Convergence, strong      3
Convergence, to $\infty$      77
Convergence, to $\infty_s$      77
Convolution      156
Convolution operator      26 156
Cross diagonal      19
Diagonal      19
Diagonals of A      21
Dual space      6
Elementwise invertible      36
Essential cluster point      96
Essential range      96
Essential spectrum      52
Essentiallyinvertible      36
Factor algebra 2 final part      41
Finite section method for BC      164
Finite section method, FSM      40
Finite section projector      9
Fourier transform      156
Fredholm operator      51
Fredholm's alternative      52
Gelfand transform      118
Gelfand transformation      118
Generalized multiplicator      6
Homomorphism      75
Ideal      2
Image      3
INDEX      51
Information flow      32
Integral operator      16
Integral operator, singular      130
Inverse closed      2
Invertibility at infinity      54
Invertible      2 3
Invertible at infinity      54
Invertible at infinity, weakly      54
Invertible elementwise      36
Invertible essentially      36
Invertible uniformly      36
Kernel      3
Kernel function      16
Laurent operator      22
Layer      60
Limit operator      78
Local algebra      123
Local essential range      96
Local operator spectrum      79
Local oscillation      92
Local principle by Allan and Douglas      122
Local principle, main idea      76
Local representatives      123
Locallycompact      59 157
Lower norm      69
Main diagonal      19
Massive set      37
Matrix entry      16
Matrix representation      16
Maximal ideal      118
Minus-index      88
Multiplicator      6
Neighbourhood of $\infty$      77
Neighbourhood of $\infty_s$      77
Noise      111
Norm convergence      3
Operator norm      3
Operator spectrum      79
Ordinary function      93
P-convergence      42
P-Fredholm operator      53
P-limit      42
P-sequentially compact      139
Plus-index      88
Poor function      93
Pre-adjoint operator      7
Pre-dual space      6
Pseudo-ergodic      109
Pseudo-ergodic w.r.t. D      108
Pseudospectrum      89
Quasidiagonal operator      12
Reflexive      7
Relaxation      101
Rich function      93
Rich operator      80
Row      19
Schrodinger operator      132
Semi-almost periodic function      105
Semi-Fredholm operator      71
Shift invariant      81
Shift operator      6
Shifts of A      134
Singular integral operator      130
Slowly oscillating      97 100
Spectrum      2
Stable      36
Stacked operator      60
Step function      93
Stone — Cech boundary      120
Stone — Cech compactification      120
Strong convergence      3
Sufficient family      76
Sufficiently smooth      37
Toeplitz operator      129
Translation invariant      81
Uniformly bounded below      141
Uniformly invertible      36
Unit element      2
Unital homomorphism      75
Weakly invertible at infinity      54
Wiener algebra      26
Wiener — Hopf operator      129
Z-almost periodic      104
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