### Conley index in Hilbert spaces and the Leray-Schauder degree

DOI: http://dx.doi.org/10.12775/TMNA.2009.010

#### Abstract

Let $H$ be a real infinite dimensional and separable Hilbert space.

With an isolated invariant set $\inv(N)$ of a flow $\phi^t$ generated by an

$\mathcal L\mathcal S$-vector field $f\:H\supseteq \Omega\to H$, $f(x)=Lx+K(x)$,

where $L\:H\to H$ is strongly indefinite linear operator and

$K\:H\supseteq \Omega\to H$

is completely continuous, one can associate a homotopy invariant $h_{\mathcal L\mathcal S}(\inv(N),\phi^t)$

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

$\deg_{\mathcal L\mathcal S}(\widehat{f},N,0)$, where $\widehat f:H\supseteq \Omega\to H$ is defined as $\widehat

f(x)=x+L^{-1}K(x)$.

With an isolated invariant set $\inv(N)$ of a flow $\phi^t$ generated by an

$\mathcal L\mathcal S$-vector field $f\:H\supseteq \Omega\to H$, $f(x)=Lx+K(x)$,

where $L\:H\to H$ is strongly indefinite linear operator and

$K\:H\supseteq \Omega\to H$

is completely continuous, one can associate a homotopy invariant $h_{\mathcal L\mathcal S}(\inv(N),\phi^t)$

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

$\deg_{\mathcal L\mathcal S}(\widehat{f},N,0)$, where $\widehat f:H\supseteq \Omega\to H$ is defined as $\widehat

f(x)=x+L^{-1}K(x)$.

#### Keywords

Conley index in Hilbert spaces; Leray-Schauder degree; LS-vector field; strongly indefinite operator

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