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

Abstract

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.