We consider the nonlinear eigenvalue problem where 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,) SR R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p= (x, 0, ) of the unperturbed problem, we prove that the connected component of containing pis either unbounded or meets a triple p= (x, 0,) with p6= p. When C is the identity and G = H is finite dimensional, the assumptions on (x, 0,) mean that xis an eigenvector of L 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.

### Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces

#### Abstract

We consider the nonlinear eigenvalue problem where 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,) SR R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p= (x, 0, ) of the unperturbed problem, we prove that the connected component of containing pis either unbounded or meets a triple p= (x, 0,) with p6= p. When C is the identity and G = H is finite dimensional, the assumptions on (x, 0,) mean that xis an eigenvector of L 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2020
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11566/290093`
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

##### Citazioni
• ND
• 3
• 3