We consider the nonlinear eigenvalue problem
where are real parameters, are bounded linear operators between separable real Hilbert spaces, and is a continuous map defined on the unit sphere of . We prove a global persistence result regarding the set of the solutions of this problem. Namely, if the operators and are compact, under suitable assumptions on a solution of the unperturbed problem, we prove that the connected component of containing is either unbounded or meets a triple with . When is the identity and is finite dimensional, the assumptions on mean that is an eigenvector of whose corresponding eigenvalue 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.
Cite this article
Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera, Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces. Z. Anal. Anwend. 39 (2020), no. 4, pp. 475–497DOI 10.4171/ZAA/1669