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...

The aim of the present book is to propose a new algebraic approach to the study of norm stability of operator sequences which arise, for example, via discretization of singular integral equations on composed curves. A wide variety of discretization methods, including quadrature rules and spline or wavelet approximations, is covered and studied from a unique point of view. The approach takes advantage of the fruitful interplay between approximation theory, concrete operator theory, and local Banach algebra techniques. The book is addressed to a wide audience, in particular to mathematicians...

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.