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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Parity and generalized multiplicity

Authors: P. M. Fitzpatrick and Jacobo Pejsachowicz
Journal: Trans. Amer. Math. Soc. 326 (1991), 281-305
MSC: Primary 58E07; Secondary 47H15, 58C99
MathSciNet review: 1030507
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Abstract: Assuming that $ X$ and $ Y$ are Banach spaces and $ \alpha :[a,b] \to \mathcal{L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $ [{\text{F}} - \text{P}1]$ we defined a homotopy invariant of $ \alpha,\sigma (\alpha,I) \in {{\mathbf{Z}}_2}$, the parity of $ \alpha $ on $ I$. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a $ \bmod\, 2$ count of the number of transversal intersections of $ \alpha (I)$ with the set of singular operators, (b) that if $ {\lambda _0}$ is an isolated singular point of $ \alpha $, then the local parity

$\displaystyle \sigma (\alpha,{\lambda _0}) \equiv \mathop {\lim }\limits_{\vare... ...\to 0} \sigma (\alpha,[{\lambda _0} - \varepsilon,{\lambda _0} + \varepsilon ])$

remains invariant under Lyapunov-Schmidt reduction, and (c) that $ \sigma (\alpha,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}$, where $ {M_G}({\lambda _0})$ is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.

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PII: S 0002-9947(1991)1030507-7
Article copyright: © Copyright 1991 American Mathematical Society