TOPOLOGICAL PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED LINEAR OPERATOR RAFFAELE CHIAPPINELLI, MASSIMO FURI, AND MARIA PATRIZIA PERA Abstract. Let T be a selfadjoint 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 xε ∈ S and λε near λ 0 , such that Txε + 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 λ 0 is odd cannot be removed. 1. Introduction Let T be a selfadjoint bounded operator acting in a real Hilbert space H, and denote by S the unit sphere of H. Let x 0 be a unit λ 0 -eigenvector of T , i.e. x 0 belongs to S and is an eigenvector of T with eigenvalue λ 0 ∈ R. Given a (possibly nonlinear) continuous operator B : X → X, consider the perturbed “eigenvalue” problem Tx + εB(x)= λx, x ∈ S. (1.1) We say that x 0 is continuable for problem (1.1) if there exists a continuous function ε → (λ ε ,x ε ) of an interval (−δ,δ) into R × X such that Tx ε + εB(x ε )= λ ε x ε , ∀ε ∈ (−δ,δ). Notice that a unit λ 0 -eigenvector x 0 of T is continuable if it is also an eigenvector of B; meaning, as in the linear case, that B(x 0 )= µx 0 for some µ ∈ R. Indeed, ε → (λ 0 + εµ,x 0 ) is the required continuous function. For a simple example of a perturbed eigenvalue problem without any continuable eigenvector (but with nonempty sphere of unit eigenvectors), take H = R 2 , T the zero operator, and B :(x,y) → (−y,x). In [2] the first author proved that if λ 0 is an isolated simple eigenvalue of T , then, under the mild assumption that B is Lipschitz continuous, each of the two unit λ 0 -eigenvectors is continuable (in a Lipschitz continuous fashion). In [3] we considered the case in which λ 0 is an isolated eigenvalue of T of multi- plicity greater than 1 (algebraic and geometric multiplicities coincide in the selfad- joint case) and we gave necessary, as well as sufficient, conditions for a given unit λ 0 -eigenvector to be continuable. In the same article we formulated the conjecture that when the multiplicity of λ 0 is odd, then, whatever is the nonlinear operator B (provided it is continuous), the even dimensional sphere of unit λ 0 -eigenvectors contains at least one vector x 0 which is “persistent” in the following sense: there Date : August 5, 2014. 2000 Mathematics Subject Classification. Primary 47H14; Secondary 47A55, 47J10, 47J15. 1