Global perturbation of nonlinear eigenvalues (Q2041944)

Let \(U\) and \(V\) be two Banach spaces and \(\mathcal{L}(\lambda,\mu):U\to V\) be a Fredholm operator of index \(0\) for any \((\lambda,\mu)\in [a,b]\times [c,d]\). Here \(\lambda\) is the spectral parameter on which \(\mathcal{L}\) depends holomorphically, as well as nonlinearly; while \(\mu\) is the perturbation parameter on which \(\mathcal{L}\) depends only continuously. This paper generalizes the classical theory of perturbation of eigenvalues to the operator surface \(\mathcal{L}\) satisfying the above listed hypotheses. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [\textit{T. Kato}, Perturbation theory for linear operators. Reprint of the corr. print. of the 2nd ed. 1980. Berlin: Springer-Verlag (1995; Zbl 0836.47009), Chapter 2, Section 5]).