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 is an isolated eigenvalue of T of odd multiplicity greater than 1. Given an arbitrary operator B:H ! H of class C1, we prove that for any sufficiently small there exists such that Tx" C "B.x". 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 0 is odd cannot be removed.