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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.

Аннотация:

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.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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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,       42 Convergence, *-strong      10 Convergence, K-strong      10 Convergence, norm-      3 Convergence, pre*-strong      10 Convergence, strong      3 Convergence, to       77 Convergence, to       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       77 Neighbourhood of       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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