| dc.description.abstract | Theorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on R-n. In the paper we consider finite dimensional inner product spaces (E,Phi) over a field K = F((chi(1),...,chi(m))) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, Phi) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, Phi) -> (E, Phi) is decomposable except when dim E is a power of 2 with exponent at most m, and Phi is a tenser product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators. (C) 2008 Mathematical Institute Slovak Academy of Sciences. | |
