Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator

✍ Scribed by Raffaele Chiappinelli; Massimo Furi; Maria Patrizia Pera

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 370 KB
• Volume: 23
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2009.09.011

✦ Synopsis

S the unit sphere of H. Assume that λ 0 is an isolated eigenvalue of T of odd multiplicity greater than 1. Given an arbitrary operator B:H → H of class C 1 , we prove that for any ε = 0 sufficiently small there exists x ε ∈ S and λ ε near λ 0 , such that Tx ε + εB(x ε ) = λ ε x ε . This result was conjectured, but not proved, in a previous article by the authors.

We provide an example showing that the assumption that the multiplicity of λ 0 is odd cannot be removed.