Topological persistence of the unit eigenvectors of a perturbed Fredholm operator of index zero

Let $A,Ccln E o F$ be two bounded linear operators between real Banach spaces, and denote by $S$ the unit sphere of $E$ (or, more generally, let $S = gsp{-1}(1)$, where $g$ is any continuous norm in $E$). Assume that $mu_0$ is an eigenvalue of the problem $Ax = mu Cx$, that the operator $L = A - mu_0 C$ is Fredholm of index zero, and that $C$ satisfies the transversality condition $Im L + C(Ker L) = F$, which implies that the eigenvalue $mu_0$ is isolated (and when $F=E$ and $C$ is the identity implies that the geometric and the algebraic multiplicities of $mu_0$ coincide). We prove the following result about the persistence of the unit eigenvectors: emph{Given an arbitrary $C^1$ map $M cln E o F$, if the (geometric) multiplicity of $mu_0$ is odd, then for any real $e$ sufficiently small there exists $x_e in S$ and $mu_e$ near $mu_0$ such that $ Ax_e + e M(x_e) = mu_e Cx_e. $} This result extends a previous one by the authors in which $E$ is a real Hilbert space, $F=E$, $A$ is selfadjoint and $C$ is the identity. We provide an example showing that the assumption that the multiplicity of $mu_0$ is odd cannot be removed.