  We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = \lambda
Cx$, $\|x\|=1$, where $\varepsilon,\lambda$ are real parameters, $L, C\colon G
\to H$ are bounded linear operators between separable real Hilbert spaces, and
$N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We
prove a global persistence result regarding the set $\Sigma$ of the solutions
$(x,\varepsilon,\lambda) \in S \times \mathbb R\times \mathbb R$ of this
problem. Namely, if the operators $N$ and $C$ are compact, under suitable
assumptions on a solution $p_*=(x_*,0,\lambda_*)$ of the unperturbed problem,
we prove that the connected component of $\Sigma$ containing $p_*$ is either
unbounded or meets a triple $p^*=(x^*,0,\lambda^*)$ with $p^* \not= p_*$. When
$C$ is the identity and $G=H$ is finite dimensional, the assumptions on
$(x_*,0,\lambda_*)$ mean that $x_*$ is an eigenvector of $L$ whose
corresponding eigenvalue $\lambda_*$ is simple. Therefore, we extend a previous
result obtained by the authors in the finite dimensional setting. Our work is
inspired by a paper of R. Chiappinelli concerning the local persistence
property of the unit eigenvectors of perturbed self-adjoint operators in a real
Hilbert space.
