POSITIVE OPERATORS

1.2. Extensions of Positive Operators 29
Theorem 1.32 (Kantorovich). Let E and F be two ordered vector spaces
with F a Dedekind complete Riesz space. If G is a majorizing vector subspace
of E and T : G → F is a positive operator, then T has a positive linear
extension to all of E.
Proof. Fix x ∈ E and let y ∈ G satisfy x ≤ y. SinceG is majorizing there
exists a vector u ∈ G with u ≤ x. Hence, u ≤ y and the positivity of T
implies T (u) ≤ T (y) for all y ∈ G with x ≤ y. In particular, it follows that
inf T (y): y ∈ G and x ≤ y exists in F for each x ∈ E. Thus, a mapping
p: E → F can be defined via the formula
p(x) =inf T (y): y ∈ G and x ≤ y .
Clearly, T (x) =p(x) holds for each x ∈ G and an easy argument shows that
p is also sublinear.
Now, by the Hahn–Banach Extension Theorem 1.25, the operator T has
a linear extension S to all of E satisfying S(z) ≤ p(z) foreachz ∈ E. If
z ∈ E + , then −z ≤ 0, and so from
−S(z) =S(−z) ≤ p(−z) ≤ T (0) = 0 ,
we see that S(z) ≥ 0. This shows that S is a positive linear extension of T
to all of E.
It is a remarkable fact that in case the domain of a positive operator T
is a majorizing vector subspace, then the convex set E(T ) is not merely nonempty
but it also has extreme points. This result is due to Z. Lipecki [115].
Theorem 1.33 (Lipecki). Let E and F be two Riesz spaces with F Dedekind
complete. If G is a majorizing vector subspace of E and T : G → F is a
positive operator, then the nonempty convex set E(T ) has an extreme point.
Proof. According to Theorem 1.31 we must establish the existence of some
S ∈E(T ) satisfying
for all x ∈ E.
inf S |x − y| : y ∈ G =0
Start by considering pairs (H, S) where H is a vector subspace majorizing
E and S : H → F is a positive operator. For every such pair (H, S)
define p H,S : E → F by
p H,S (x) =inf S(y): y ∈ H and x ≤ y .
It should be clear that p H,S is a sublinear mapping satisfying p H,S (y) =S(y)
for every y ∈ H. In addition, if (H1,S1) and(H2,S2) satisfyH1 ⊆ H2 and
S2 = S1 on H1, then p H2 ,S 2 (x) ≤ p H1 ,S 1 (x) holds for all x ∈ E.
Now let C be the collection of all pairs (H, S) such that: