Let A; C : E -> 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(-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 C-0 is Fredholm of index zero, and that C satis fi es 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 : E -> F, if the (geometric) multiplicity of mu(0) is odd, then for any real epsilon sufficiently small there exists x(epsilon) is an element of S and mu(epsilon) near mu(0) such that Ax(epsilon) + epsilon M (x(epsilon)) = mu(epsilon)Cx(epsilon). 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.

Chiappinelli, R., Furi, M., Pera, M.P. (2014). Topological persistence of the unit eigenvectors of a perturbed Fredholm operator of index zero. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 33(3), 347-367 [10.4171/ZAA/1516].

Topological persistence of the unit eigenvectors of a perturbed Fredholm operator of index zero

CHIAPPINELLI, RAFFAELE;
2014-01-01

Abstract

Let A; C : E -> 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(-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 C-0 is Fredholm of index zero, and that C satis fi es 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 : E -> F, if the (geometric) multiplicity of mu(0) is odd, then for any real epsilon sufficiently small there exists x(epsilon) is an element of S and mu(epsilon) near mu(0) such that Ax(epsilon) + epsilon M (x(epsilon)) = mu(epsilon)Cx(epsilon). 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.
2014
Chiappinelli, R., Furi, M., Pera, M.P. (2014). Topological persistence of the unit eigenvectors of a perturbed Fredholm operator of index zero. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 33(3), 347-367 [10.4171/ZAA/1516].
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/11365/986887
 Attenzione

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