Compact perturbations of Fredholm operators on Norm Hilbert spaces over Krull valued fields

A continuous linear operator of a Banach space into itself is called Fredholm if its kernel and cokernel are finite-dimensional. The subject of compact perturbations of Fredholm operators on complex spaces is well-known, see e.g. [6]. For spaces over non-archimedean valued fields K of rank 1 (i.e. the range of the valuation is in [0, infinity)) the preservation of Fredholm operators under compact perturbations was proved in [1]. In this paper we allow K to have an arbitrary totally ordered abelian value group G (rather than a subgroup of (0, infinity)), but we restrict our study to Norm Hilbert Spaces (NHS) E over K i.e. each closed subspace admits a projection of norm <= 1. We prove the striking fact that the index of a Fredholm operator is 0. Further, we consider a natural class phi(E) of so-called Lipschitz-Fredholm operators and prove that the operators A for which A + phi(E) subset of phi(E) form precisely the set of all nuclear operators. (An operator A is called nuclear if there exists a sequence A(1), A(2), ... of continuous finite rank operators such that parallel to Ax - A(n)x parallel to < gn parallel to x parallel to (x is an element of E\ {0}) for some sequence g(1), g(2), ... in G, tending to 0. Here the strict inequality is essential!)