JournalszaaVol. 33, No. 3pp. 347–367

Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero

  • Raffaele Chiappinelli

    Università degli Studi di Siena, Italy
  • Massimo Furi

    Università di Firenze, Italy
  • Maria Patrizia Pera

    Universita di Firenze, Italy
Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero cover
Download PDF

Abstract

Let A,C ⁣:EFA,C\colon E \to F be two bounded linear operators between real Banach spaces, and denote by SS the unit sphere of EE (or, more generally, let S=g\sp1(1)S = g\sp{-1}(1), where gg is any continuous norm in EE). Assume that μ0\mu_0 is an eigenvalue of the problem Ax=μCxAx = \mu Cx, that the operator L=Aμ0CL = A - \mu_0 C is Fredholm of index zero, and that CC satisfies the transversality condition \ImgL+C(\KerL)=F\Img L + C(\Ker L) = F, which implies that the eigenvalue μ0\mu_0 is isolated (and when F=EF=E and CC is the identity implies that the geometric and the algebraic multiplicities of μ0\mu_0 coincide). We prove the following result about the persistence of the unit eigenvectors: Given an arbitrary C1C^1 map M ⁣:EFM \colon E \to F, if the (geometric) multiplicity of μ0\mu_0 is odd, then for any real ε\varepsilon sufficiently small there exists xεSx_\varepsilon \in S and με\mu_\varepsilon near μ0\mu_0 such that \lb Axε+εM(xε)=μεCxε.Ax_\varepsilon + \varepsilon M(x_\varepsilon) = \mu_\varepsilon Cx_\varepsilon. This result extends a previous one by the authors in which EE is a real Hilbert space, F=EF=E, AA is selfadjoint and CC is the identity. We provide an example showing that the assumption that the multiplicity of μ0\mu_0 is odd cannot be removed.

Cite this article

Raffaele Chiappinelli, Massimo Furi, Maria Patrizia Pera, Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero. Z. Anal. Anwend. 33 (2014), no. 3, pp. 347–367

DOI 10.4171/ZAA/1516