Global Perturbation of Nonlinear Eigenvalues

Abstract

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){\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)}, (λ,μ)↦𝔏⁢(λ,μ){(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)}, depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ0⁢(U,V){\Phi_{0}(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]).

Keywords

• eigenvalue perturbation
• spectral parameter
• perturbation parameter
• generalized algebraic multiplicity
• nonlinear spectral theory
• fredholm operators
• intricate weighted eigenvalue problems
• 47a55
• 47h11
• 35j25