Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero

Abstract

Let $A,C:E\to F$ 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 $L=A-{\mu }_{0}C$ is Fredholm of index zero, and that $C$ satisfies the transversality condition $\mathrm{Img}L+C(\mathrm{Ker}L)=F$, 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 $M:E\to F$, if the (geometric) multiplicity of ${\mu }_0$ is odd, then for any real $\epsilon$ sufficiently small there exists $x_{\epsilon }\in S$ and ${\mu }_{\epsilon }$ near ${\mu }_0$ such that $A{x}_{\epsilon }+\epsilon M(x_{\epsilon })={\mu }_{\epsilon }C{x}_{\epsilon }.$

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.