AbstractThis paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface𝔏:[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 theperturbation parameter, μ, and holomorphically, as well as nonlinearly, on thespectral parameter, λ, whereΦ0⁢(U,V){\Phi_{0}(U,V)}stands for the set of Fredholm operators of index zero betweenUandV. 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]).