POSITIVE OPERATORS

1.1. Basic Properties of Positive Operators 19
In particular, we have n m i=1 j=1 zij = x. On the other hand, using the
lattice identity x ∨ y = 1
2 (x + y + |x − y|), we see that
Similarly,
n
S(xi) ∨ T (xi)
i=1
= 1
2
= 1
2
≤ 1
2
=
n
S(xi)+T (xi)+|S(xi) − T (xi)|
i=1
n m S(zij)+
i=1
j=1
m m
T (zij)+ S(zij) − T (zij)
j=1
j=1
n m
S(zij)+T (zij)+|S(zij) − T (zij)|
i=1
n
i=1 j=1
j=1
m
S(zij) ∨ T (zij) .
m
S(yj) ∨ T (yj) ≤
j=1
holds, and so D is directed upward.
n
i=1 j=1
m
S(zij) ∨ T (zij)
(2) Use (1) in conjunction with the identity T ∧ S = − (−S) ∨ (−T ) .
(3) Use (1) and the identity |T | = T ∨ (−T ).
The next result presents an interesting local approximation property of
positive operators.
Theorem 1.22. Let T : E → F be a positive operator between two Riesz
spaces with F Dedekind σ-complete. Then for each x ∈ E + there exists a
positive operator S : E → F such that:
(1) 0 ≤ S ≤ T .
(2) S(x) =T (x).
(3) S(y) =0for all y ⊥ x.
Proof. Let x ∈ E + be fixed and define S : E + → F + by
S(y) =sup T (y ∧ nx): n =1, 2,... .
(The supremum exists since F is Dedekind σ-complete and the sequence
{T (y ∧ nx)} is bounded above in F by Ty.) We claim that S is additive.
To see this, let y, z ∈ E + .From(y + z) ∧ nx ≤ y ∧ nx + z ∧ nx we get
T (y + z) ∧ nx ≤ T (y ∧ nx)+T (z ∧ nx) ≤ S(y)+S(z) ,