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

We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = \lambda Cx$,$|x|=1$,where$\varepsilon,\lambda$arerealparameters,$L, C\colon G \to H$areboundedlinearoperatorsbetweenseparablerealHilbertspaces,and$N\colon S \to H$isacontinuousmapdefinedontheunitsphereof$G$. We prove a global persistence result regarding the set $\mathrm{\Sigma }$ of the solutions $(x,\epsilon ,\lambda )\in S\times \mathbb{R}\times \mathbb{R}$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_{\ast }=(x_{\ast },0,{\lambda }_{\ast })$ of the unperturbed problem, we prove that the connected component of $\mathrm{\Sigma }$ containing $p_{\ast }$ is either unbounded or meets a triple $p^{\ast }=(x^{\ast },0,{\lambda }^{\ast })$ with $p^{\ast }≠p_{\ast }$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_{\ast },0,{\lambda }_{\ast })$ mean that $x_{\ast }$ is an eigenvector of $L$ whose corresponding eigenvalue ${\lambda }_{\ast }$ 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.