Conley index in Hilbert spaces and the Leray-Schauder degree

Abstract

Let $H$ be a real infinite dimensional and separable Hilbert space. With an isolated invariant set ${\rm inv}(N)$ of a flow $\phi^t$ generated by an $\mathcal L\mathcal S$-vector field $f\colon H\supseteq \Omega\to H$, $f(x)=Lx+K(x)$, where $L\colon H\to H$ is strongly indefinite linear operator and $K\colon H\supseteq \Omega\to H$ is completely continuous, one can associate a homotopy invariant $h_{\mathcal L\mathcal S}({\rm inv}(N),\phi^t)$ called the $\mathcal L\mathcal S$-Conley index. In fact, this is a homotopy type of a finite CW-complex. We define the Betti numbers and hence the Euler characteristic of such index and prove the formula relating these numbers to the Leray-Schauder degree ${\rm deg}_{\mathcal L\mathcal S}(\widehat{f},N,0)$, where $\widehat f\colon H\supseteq \Omega\to H$ is defined as $\widehat f(x)=x+L^{-1}K(x)$.