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- Title
Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces.
- Authors
Benevieri, Pierluigi; Calamai, Alessandro; Furi, Massimo; Pera, Maria Patrizia
- Abstract
We consider the nonlinear eigenvalue problem Lx + N(x) = λCx; ||x|| = 1; where λ are real parameters, L;C : G ! H are bounded linear operators between separable real Hilbert spaces, and N : S ! H is a continuous map defined on the unit sphere of G. We prove a global persistence result regarding the set of the solutions (x; λ) 2 S ×R×R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p* = (x*; 0; λ*) of the unperturbed problem, we prove that the connected component of containing p* is either unbounded or meets a triple p* = (x*; 0; λ*) with p* 6= p*. When C is the identity and G = H is finite dimensional, the assumptions on (x*; 0; λ*) mean that x* is an eigenvector of L whose corresponding eigenvalue λ* 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.
- Subjects
HILBERT space; EIGENVALUES; LINEAR operators; FREDHOLM operators; NONLINEAR theories; EIGENVECTORS
- Publication
Journal of Analysis & Its Applications / Zeitschrift für Analysis & ihre Anwendungen, 2020, Vol 39, Issue 4, p475
- ISSN
0232-2064
- Publication type
Article
- DOI
10.4171/ZAA/1669