# Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces

### Pierluigi Benevieri

Universidade de São Paulo, Brazil### Alessandro Calamai

Università Politecnica delle Marche, Ancona, Italy### Massimo Furi

Università degli Studi di Firenze, Italy### Maria Patrizia Pera

Università degli Studi di Firenze, Italy

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## Abstract

We consider the nonlinear eigenvalue problem

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.

## 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. 475–497

DOI 10.4171/ZAA/1669