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The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators on/P-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the generM theory, we present its specifications to concrete classes of band-dominated operators. IntroductionThe topic of the present paper is band operators and norm limits of band operators -the latter we call band-dominated operators. Let 12(2Z) denote the ttilbert space of the two-sided infinite and squared-summable sequences provided with its standard basis It is easy to check that every band operator can be uniquely written as a finite sum ~tr a~V; where the a~ are bounded multiplication operators (i.e. they are given by a diagonal matrix with respect to the standard basis), and where the V; are the shift operators ej ~-+ ei+j. Conversely, every finite sum ~ aiVi is a band operator. This equivalence allows to think of band operators as being composed of two kinds of 'generators' -multiplication and shift operators -and we shall adopt this point of view in what follows.Band and band-dominated operators appear in numerous branches of mathematics. Archetypical appearances are that as discretizations of partial differential or pseudo-differential operators, and that in signal processing. We shall examine these operators with *Supported by the DFG grant 436 RUS/17/148/95 ?Supported by a DFG Heisenberg grant

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