Global Perturbation of Nonlinear Eigenvalues

This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface L: [a, b] × [c, d] → Φ0(U, V), (λ, μ) → L(λ, μ), depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ0(U, V) stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).