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In this book we are concerned with the study of a certain class of infinite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their finite 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 infinite matrices as bounded linear operators on a Banach space E of two-sided infinite sequences.The class of operators we are interested in consists of those bounded and linear operatorson E which can be approximated in the operator norm by b and matrices. 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