Let A(t) be a path of bounded operators on a real Hilbert space, hyperbolic at ±∞. 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 = 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.

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In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle -or more generally an essential subbundle -of the tangent bundle of M , suitably related with the gradient flow of f . This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W , possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W , classification, up to topological conjugacy, of the gradient flow on the closure of W , in the case dim W = 2. 1 The two facts would actually be equivalent if we were using coefficients in a field, instead of the ring Z. 2 Here one needs just that the unstable manifold W u (x) and the stable manifold W s (y) have empty intersection, for every pair of distinct critical points x, y with m(x) ≤ m(y).

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