This chapter presents a concise introduction to the abstract theory of Fredholm operators. It also contains some examples; however the main applications will concern Wiener-Hopf integral operators and Toeplitz operators which we shall deal with in the next chapters and in Volume II. The first section contains the definition of a Fredholm operator and the first examples. In Section 2 we pay special attention to operators with closed range. The basic perturbation theorems and the properties of the index are given in Sections 3 and 4. In Section 5 Fredholm operators are studied in the framework of the Calkin algebra. Connections with generalized invertibility appear in Section 6. Index formulas in terms of trace and determinant are given in Section 7. Sections 8 and 9 are devoted to Fredholm operator valued functions that are analytic and to equivalence of such functions. An operator theory generalization of Rouché’s theorem appears here. The last section concerns singular values of bounded operators and their connections with the essential spectrum.
KeywordsOperator Function Compact Operator Toeplitz Operator Finite Type Essential Spectrum
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