Zur Stabilität der algebraischen vielfachheit von eigenwerten von holomorphen fredholm-operatorfunktionen

In this paper we are concerned with the stability of the algebraic multiplicity of eigenvalues of holomorphic operator-valued functions in the framework of a generalized perturbation theory which is at the same time well-suited for the treatment of approximation methods. It is shown that the algebraic multiplicity of an isolated eigenvalue is stable if the operators under consideration are restricted to the class of so-called approximation-proper families of holomorphic functions of Fredholm mappings acting in certain discrete approximations. As an essential tool in this context the representation formula for the algebraic multiplicity of A. S. Markus and E. I. Sigal is used. The above problem has been rigorously studied in the case when the eigenvalue parameter occurs linearly, but even in this case our result is of interest.