  We consider abstract equations of the form Ax=-z on a locally convex space,
where A generates a positive semigroup and z is a positive element. This is an
abstract version of the operator Lyapunov equation A*P+PA=-Q from control
theory. It is proved that under suitable assumptions existence of a positive
solution implies that -A has a positive inverse, and the generated semigroup is
asymptotically stable. We do not require that z is an order unit, or that the
space contains any order units. As an application, we generalize Wonham's
theorem on the operator Lyapunov equations with detectable right hand sides to
reflexive Banach spaces.
