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

Let A; C : E -> F be two bounded linear operators between real Banach spaces, and denote by S the unit sphere of E (or, more generally, let S = g(-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 C-0 is Fredholm of index zero, and that C satis fi es the transversality condition Img 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: Given an arbitrary C-1 map M : E -> F, if the (geometric) multiplicity of mu(0) is odd, then for any real epsilon sufficiently small there exists x(epsilon) is an element of S and mu(epsilon) near mu(0) such that Ax(epsilon) + epsilon M (x(epsilon)) = mu(epsilon)Cx(epsilon). 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.