Endomorphisms of Partially Ordered Vector Spaces

A vector space V over the real field R is said to be partially ordered if a non-empty subset F + is specified which satisfies the following axioms: (i) if x and y are in V and oc 0, then x + y and VLX are in V) (ii) if x and — x are in T then x = 0. We write, as usual, x ^ y (or y 0 is an or^er MWI if for each x e V there exists f e R with fs ^ #; for such an e we define the functional pe by p6(x) = inf [f e R : fs ^ #]. A linear transformation T of V into itself is called an endomorphism if 7# ^ 0 whenever # ^ 0. A positive linear functional is a non-zero linear functional cp such that 99 (#) ^ 0 whenever x ^ 0. We prove the following theorem. Let V be a partially ordered vector space with an order unit e and let A be an endomorphism of V. Then there exists a positive linear functional (p with i_ A*cp = Qcp [i.e. with cp(Ax) = £9?(#) for all # e F], where £ = lim {pe(A e)}n. n->oo A theorem of this type, but less general and without the determination of Q, has been proved by M. G. Krein using the fixed point theorem of J. Schauder. Our proof does not use any fixed point theorem but depends on some elementary results in the theory of Banach algebras. Many related results have been proved by M. G. Krein and M. A. Rutman [Uspehi Matem. Nauk (N.S.) 3, no 1 (23), 3—95 (1948)]. Their principal results on linear operators leaving invariant a cone with interior may be deduced from our theorem with some gain in simplicity and generality. For example, if V is a reflexive Banach space and F + is a closed cone K such that K and K* (the set of non-negative linear functionals) have interior points, and if A is an endomorphism with spectral radius Q, then there are non-zero vectors ueK, cpeK* with Au = QU and A*cp = QCp.