Spectral decompositions of operators on non-Archimedean orthomodular spaces

The most central property of an infinite-dimensional Hilbert space is expressed by the projection theorem: Every orthogonally closed linear subspace is an orthogonal summand. Besides the obvious Hilbert spaces, there exist other infinite-dimensional orthomodular spaces. Here we study bounded linear operators on an orthomodular spaceE constructed over a field of generalized power series with real coefficients. Our main result states that every bounded, self-adjoint operator gives rise to a representation ofE as the closure of an infinite orthogonal sum of invariant subspaces each of which is of dimension 1 or 2. The proof combines the technique of reduction modulo the residual spaces with theorems on orthogonal decompositions of finite matrices over fields of power series.