Ordinary differential operators in Hilbert spaces and Fredholm pairs

Abstract.

Let $A(t)$ be a path of bounded operators on a real Hilbert space, hyperbolic at $\pm \infty$. We study the Fredholm theory of the operator $F_A=d/dt-A(t)$. We relate the Fredholm property of $F_A$ to the stable and unstable linear spaces of the associated system $X^{\prime}=A(t)X$. Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds.