Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator

Let TT be a self-adjoint bounded operator acting in a real Hilbert space HH, and denote by SS the unit sphere of HH. Assume that λ0λ0 is an isolated eigenvalue of TT of odd multiplicity greater than 11. Given an arbitrary operator B:H→H of class C1C1, we prove that for any ε≠0ε≠0 sufficiently small there exists xε∈Sxε∈S and λελε near λ0λ0, such that Txε+εB(xε)=λεxεTxε+εB(xε)=λεxε. This result was conjectured, but not proved, in a previous article by the authors.We provide an example showing that the assumption that the multiplicity of λ0λ0 is odd cannot be removed.