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This text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = 2(Z) be the Hilbert space of all squared summable functions x: Z -] Xi provided with the norm 2 2 X IIxl1: =L I iI . iEZ It is often convenient to think of the elements x of 2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of 2(Z) is the family of sequences (ei)iEZ where ei = (. . .,0,0, 1,0,0, . . . ) with the 1 standing at the ith place. Every bounded linear operator A...

This is the first monograph devoted to a fairly wide class of operators, namely band and band-dominated operators and their Fredholm theory. The main tool in studying this topic is limit operators. Applications are presented to several important classes of such operators: convolution type operators and pseudo-differential operators on bad domains and with bad coefficients.