Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces

We consider the nonlinear eigenvalue problem $Lx + e N(x) = l Cx$, $|x|=1$, where $e,l$ are real parameters, $L, C$ are bounded linear operators between separable real Hilbert spaces G and H, and $N$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $Sigma$ of the solutions $(x,e,l) in SxRxR$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_*=(x_*,0,l_*)$ of the unperturbed problem, we prove that the connected component of $Sigma$ containing $p_*$ is either unbounded or meets a triple $p*=(x*,0,l*)$ with $p* $ different from $ p_*$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_*,0,l_*)$ mean that $x_*$ is an eigenvector of $L$ whose corresponding eigenvalue $l_*$ 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.