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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 3, 2014, pp. 347–367
DOI: 10.4171/ZAA/1516

Published online: 2014-07-02

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

Raffaele Chiappinelli[1], Massimo Furi[2] and Maria Patrizia Pera[3]

(1) Università degli Studi di Siena, Italy
(2) Università di Firenze, Italy
(3) Universita di Firenze, Italy

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

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

Chiappinelli Raffaele, Furi Massimo, Pera Maria Patrizia: Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero. Z. Anal. Anwend. 33 (2014), 347-367. doi: 10.4171/ZAA/1516