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.