Conley index in Hilbert spaces and the Leray-Schauder degree (Q1029927)

Let \(H\) be a Hilbert space and let \(f: H\to H\) be of the form \(f(x)=Lx+K(x)\), where \(L\) is a strongly indefinite linear bounded and invertible operator, and \(K\) is a completely continuous \(C^1\)-map. Let \(\phi^t\) denote the local flow generated by the equation \(\dot x=-f(x)\). Under some additional assumptions on \(L\), to an isolating neighborhood \(N\) for \(\phi^t\) one can associate the Conley index of the invariant part \(\mathrm{Inv}(N)\) of \(N\), denoted by \(h(\mathrm{Inv}(N),\phi^t)\), being a spectrum of some pointed topological spaces. Moreover, to \(N\) one can associate the integer number \(\mathrm{deg}_L(f,\mathrm{int}(N))\) as the Leray-Schauder degree of zeros of \(I+L^{-1}K\) in the interior of \(N\). The main theorem asserts \(\chi(h(\mathrm{Inv}(N),\phi^t))=\mathrm{deg}_L(f,\mathrm{int}(N))\), where \(\chi\) denotes the Euler characteristic.