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In this book we are concerned with the study of a certain class of in?nite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their ?nite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We think of our in?nite matrices as bounded linear operators on a Banach space E of two-sided in?nite sequences. Probably the simplest case to start with 2 +? is the space E = of all complex-valued sequences u=(u ) for which m m=?? 2 |u | is summable over m? Z. m Theclassofoperatorsweareinterestedinconsistsofthoseboundedandlinear operatorsonE whichcanbeapproximatedintheoperatornormbybandmatrices. We refer to them as band-dominated operators. Of course, these considerations 2 are not limited to the space E = . We will widen the selection of the underlying space E in three directions: p • We pass to the classical sequence spaces with 1? p??. n • Our elements u=(u )? E have indices m? Z rather than just m? Z. m • We allow values u in an arbitrary ?xed Banach spaceX rather than C.

Infinite matrices. --- Linear operators. --- Numerical analysis. --- Integral equations. --- Matrices infinies --- Opérateurs linéaires --- Analyse numérique --- Equations intégrales --- Infinite matrices. » More like this Linear operators. » More like this Numerical analysis. » More like this Integral equations. --- Infinite matrices --- Linear operators --- Numerical analysis --- Integral equations --- Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- 517.98 --- Functional analysis and operator theory --- 517.98 Functional analysis and operator theory --- Equations, Integral --- Matrices, Infinite --- Linear maps --- Maps, Linear --- Operators, Linear --- Mathematics. --- Matrix theory. --- Algebra. --- Functional analysis. --- Functional Analysis. --- Linear and Multilinear Algebras, Matrix Theory. --- Numerical Analysis. --- Mathematical analysis --- Operator theory --- Functional equations --- Functional analysis --- Matrices --- Functional calculus --- Calculus of variations