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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, Massimo Furi and Maria Patrizia Pera

(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