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Zeitschrift für Analysis und ihre Anwendungen


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Volume 39, Issue 4, 2020, pp. 475–497
DOI: 10.4171/ZAA/1669

Published online: 2020-10-22

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

Pierluigi Benevieri[1], Alessandro Calamai[2], Massimo Furi[3] and Maria Patrizia Pera[4]

(1) Universidade de São Paulo, Brazil
(2) Università Politecnica delle Marche, Ancona, Italy
(3) Università degli Studi di Firenze, Italy
(4) Università degli Studi di Firenze, Italy

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 $p^* \not= p_*$. 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.

Keywords: Fredholm operators, nonlinear spectral theory, eigenvalues, eigenvectors

Benevieri Pierluigi, Calamai Alessandro, Furi Massimo, Pera Maria Patrizia: Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces. Z. Anal. Anwend. 39 (2020), 475-497. doi: 10.4171/ZAA/1669