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- Title
The index bundle for selfadjoint Fredholm operators and multiparameter bifurcation for Hamiltonian systems.
- Authors
Skiba, Robert; Waterstraat, Nils
- Abstract
The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and Jänich in the sixties, and it is readily seen that, on complex Hilbert spaces, this so-called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that, for every real Hilbert space and every compact topological space X, there is a family of selfadjoint Fredholm operators parametrised by X - S1 which has a non-trivial index bundle. Further, we use this observation and a family index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.
- Subjects
FREDHOLM operators; SELFADJOINT operators; HILBERT space; TOPOLOGICAL spaces; HAMILTONIAN systems; COMPACT spaces (Topology); BIFURCATION diagrams
- Publication
Journal of Analysis & Its Applications / Zeitschrift für Analysis & ihre Anwendungen, 2022, Vol 41, Issue 3/4, p487
- ISSN
0232-2064
- Publication type
Article
- DOI
10.4171/ZAA/1718