JournalszaaVol. 39, No. 4pp. 475–497

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

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Abstract

We consider the nonlinear eigenvalue problem

Lx+εN(x)=λCx,x=1,Lx + \varepsilon N(x) = \lambda Cx, \quad \|x\|=1,

where ε,λ\varepsilon,\lambda are real parameters, L,C ⁣:GHL, C\colon G \to H are bounded linear operators between separable real Hilbert spaces, and N ⁣:SHN\colon S \to H is a continuous map defined on the unit sphere of GG. We prove a global persistence result regarding the set Σ\Sigma of the solutions (x,ε,λ)S×R×R(x,\varepsilon,\lambda) \in S \times \mathbb R\times \mathbb R of this problem. Namely, if the operators NN and CC are compact, under suitable assumptions on a solution p=(x,0,λ)p_*=(x_*,0,\lambda_*) of the unperturbed problem, we prove that the connected component of Σ\Sigma containing pp_* is either unbounded or meets a triple p=(x,0,λ)p^*=(x^*,0,\lambda^*) with ppp^* \not= p_*. When CC is the identity and G=HG=H is finite dimensional, the assumptions on (x,0,λ)(x_*,0,\lambda_*) mean that xx_* is an eigenvector of LL whose corresponding eigenvalue λ\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