POSITIVE OPERATORS

1.2. Extensions of Positive Operators 25
Theorem 1.27. Let E and F be Riesz spaces with F Dedekind complete.
If G is a Riesz subspace of E and T : G → F is a positive operator, then the
following statements are equivalent.
(1) T extends to a positive operator from E to F .
(2) T extends to an order bounded operator from E to F .
(3) There exists a monotone sublinear mapping p: E → F satisfying
T (x) ≤ p(x) for all x ∈ G.
Proof. (1) =⇒ (2) Obvious.
(2) =⇒ (3) Let S ∈Lb(E,F) satisfyS(x) =T (x) for all x ∈ G. Then
the mapping p: E → F defined by p(x) =|S|(x + ) is monotone, sublinear
and satisfies
T (x) ≤ T (x + )=S(x + ) ≤|S|(x + )=p(x)
for all x ∈ G.
(3) =⇒ (1) Let p: E → F be a monotone sublinear mapping satisfying
T (x) ≤ p(x) for all x ∈ G. Then the formula q(x) = p(x + ) defines a
sublinear mapping from E to F such that
T (x) ≤ T (x + ) ≤ p(x + )=q(x)
holds for all x ∈ G. Thus, by the Hahn–Banach Extension Theorem 1.25
there exists an extension R ∈L(E,F) ofT satisfying R(x) ≤ q(x) for all
x ∈ E. In particular, if x ∈ E + , then the relation
−R(x) =R(−x) ≤ q(−x) =p (−x) + = p(0) = 0
implies R(x) ≥ 0. That is, R is a positive linear extension of T to all of E,
and the proof is finished.
A subset A of a Riesz space is called solid whenever |x| ≤|y| and y ∈ A
imply x ∈ A. A solid vector subspace of a Riesz space is referred to as
an ideal. From the lattice identity x ∨ y = 1
2 (x + y + |x − y|), it follows
immediately that every ideal is a Riesz subspace.
The next result deals with restrictions of positive operators to ideals.
Theorem 1.28. If T : E → F is a positive operator between two Riesz
spaces with F Dedekind complete, then for every ideal A of E the formula
TA(x) =sup T (y): y ∈ A and 0 ≤ y ≤ x , x ∈ E + ,
defines a positive operator from E to F . Moreover, we have:
(a) 0 ≤ TA ≤ T .
(b) TA = T on A and TA =0on A d .
(c) If B is another ideal with A ⊆ B, then TA ≤ TB holds.