Über nichtlineare Spektral- und Störungstheorie

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.