Let (M,g) be a compact Riemannian manifold and Pg an elliptic, formally
self-adjoint, conformally covariant operator of order m acting on smooth
sections of a bundle over M. We prove that if Pg has no rigid eigenspaces
(see Definition 2.2), the set of functions f∈C∞(M,R) for which
Pefg has only simple non-zero eigenvalues is a residual set in
C∞(M,R). As a consequence we prove that if Pg has no rigid
eigenspaces for a dense set of metrics, then all non-zero eigenvalues are
simple for a residual set of metrics in the Cm-topology. We also prove that
the eigenvalues of Pg depend continuously on g in the Cm-topology,
provided Pg is strongly elliptic. As an application of our work, we show
that if Pg acts on C∞(M) (e.g. GJMS operators), its non-zero
eigenvalues are generically simple.Comment: To appear in Annales de l'Institut Fourie