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

### Raffaele Chiappinelli

Università degli Studi di Siena, Italy### Massimo Furi

Università di Firenze, Italy### Maria Patrizia Pera

Universita di Firenze, Italy

## Abstract

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 \( S = g\sp{-1}(1) \), where $g$ is any continuous norm in $E$). Assume that $μ_{0}$ is an eigenvalue of the problem $Ax=μCx$, that the operator $L=A−μ_{0}C$ is Fredholm of index zero, and that $C$ satisfies the transversality condition \( \Img L + C(\Ker L) = F \), which implies that the eigenvalue $μ_{0}$ is isolated (and when $F=E$ and $C$ is the identity implies that the geometric and the algebraic multiplicities of $μ_{0}$ 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 $μ_{0}$ is odd, then for any real $ε$ sufficiently small there exists $x_{ε}∈S$ and $μ_{ε}$ near $μ_{0}$ such that \lb $Ax_{ε}+εM(x_{ε})=μ_{ε}Cx_{ε}.$ 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 $μ_{0}$ is odd cannot be removed.

## Cite this article

Raffaele Chiappinelli, Massimo Furi, Maria Patrizia Pera, Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero. Z. Anal. Anwend. 33 (2014), no. 3, pp. 347–367

DOI 10.4171/ZAA/1516