Let be two bounded linear operators between real Banach spaces, and denote by the unit sphere of (or, more generally, let , where is any continuous norm in ). Assume that is an eigenvalue of the problem , that the operator is Fredholm of index zero, and that satisfies the transversality condition , which implies that the eigenvalue is isolated (and when and is the identity implies that the geometric and the algebraic multiplicities of coincide). We prove the following result about the persistence of the unit eigenvectors: Given an arbitrary map , if the (geometric) multiplicity of is odd, then for any real sufficiently small there exists and near such that \lb This result extends a previous one by the authors in which is a real Hilbert space, , is selfadjoint and is the identity. We provide an example showing that the assumption that the multiplicity of 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. 347–367DOI 10.4171/ZAA/1516