Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

Suppose we are given a finite-dimensional vector space V equipped with an F-rational action of a linearly algebraic group G, with F a characteristic zero field. We conjecture the following: to each vector v in V(F) there corresponds a canonical G(F)orbit of semisimple vectors of V. In the case of the adjoint action, this orbit is the G(F)-orbit of the semisimple part of v, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.