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We study the persistence of eigenvalues and eigenvectors of perturbedeigenvalue problems in Hilbert spaces. We assume that the unperturbed problemhas a nontrivial kernel of odd dimension and we prove a Rabinowitz-type globalcontinuation result. The approach is topological, based on a notion of degreefor oriented Fredholm maps of index zero between real differentiable Banachmanifolds.

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case