Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces (Q2658500)

Summary: We consider the nonlinear eigenvalue problem \[ Lx + \varepsilon N(x) = \lambda Cx, \quad \|x\|=1, \] 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 \(p^* \neq p_*\). 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.