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manuscripta mathematica

, Volume 14, Issue 2, pp 123–162 | Cite as

Über nichtlineare Spektral- und Störungstheorie

  • Wilhelm Singhof
Article

Abstract

A topology is defined on the set of closed operators between two Banach spaces. We show that the set of Lipschitzcontinuous operators is open in this topology; the relative topology on this subset is the natural one. The notion of spectrum is defined for nonlinear maps, and we prove among others the following facts: the resolvent set is open and depends upper semi-continuously on the operator; for Lipschitzcontinuous operators, the spectrum is compact; the resolvent map is continuous. Then we examine more closely the cases of Fréchet-differentiable maps and of isolated points in the spectrum.

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Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Wilhelm Singhof
    • 1
  1. 1.Mathematisches Institut der Universität zu KölnKöln 41

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