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

Let T be a self-adjoint bounded operator acting in a real Hilbert space H , and denote by S the unit sphere of H . Assume that λ 0 is an isolated eigenvalue of T of odd multiplicity greater than 1 . Given an arbitrary operator B : H → H of class C 1 , we prove that for any ε ≠ 0 sufficiently small there exists x ε ∈ S and λ ε near λ 0 , such that T x ε + ε B ( x ε ) = λ ε x ε . This result was conjectured, but not proved, in a previous article by the authors. vide an example showing that the assumption that the multiplicity of λ 0 is odd cannot be removed.