Fredholm Operators

  • Israel Gohberg
  • Seymour Goldberg
  • Marinus A. Kaashoek
Part of the Operator Theory: Advances and Applications book series (OT, volume 49)


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.


Operator Function Compact Operator Toeplitz Operator Finite Type Essential Spectrum 
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Copyright information

© Springer Basel AG 1990

Authors and Affiliations

  • Israel Gohberg
    • 1
  • Seymour Goldberg
    • 2
  • Marinus A. Kaashoek
    • 3
  1. 1.School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact SciencesTel-Aviv UniversityTel-AvivIsrael
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Faculteit Wiskunde en InformaticaVrije UniversiteitAmsterdamThe Netherlands

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