POSITIVE OPERATORS

1.1. Basic Properties of Positive Operators 5
(3) Note that
x + ∧ x − = (x + − x − ) ∧ 0+x − = x ∧ 0+x −
= −[(−x) ∨ 0] + x − = −x − + x − =0.
(a) Assume that x = y − z with y ≥ 0andz ≥ 0. From x = x + − x − ,
we get x + = x − + y − z ≤ x − + y, and so from Lemma 1.4 we get
x + = x + ∧ x + ≤ x + ∧ (x − + y) ≤ x + ∧ x − + x + ∧ y = x + ∧ y ≤ y.
Similarly, x − ≤ z.
(b) Let x = y − z with y ∧ z = 0. Then, using Theorem 1.3, we see that
x + =(y − z) ∨ 0=y ∨ z − z =(y + z − y ∧ z) − z = y. Similarly, x − = z.
We also have the following useful inequality regarding positive operators.
Lemma 1.6. If T : E → F is a positive operator between two Riesz spaces,
then for each x ∈ E we have
|Tx|≤T |x| .
Proof. If x ∈ E, then ±x ≤|x| and the positivity of T yields ±Tx ≤ T |x|,
which is equivalent to |Tx|≤T |x|.
A few more useful lattice identities are included in the next result.
Theorem 1.7. If x and y are elements in a Riesz space, then we have:
(1) x =(x− y) + + x ∧ y.
x + y + |x − y|
(2) x ∨ y = 1
2
(3) |x − y| = x ∨ y − x ∧ y.
(4) |x|∨|y| = 1
2 |x + y| + |x − y| .
(5) |x|∧|y| = 1
|x + y|−|x− y| .
2
(6) |x + y|∧|x − y| = |x|−|y| .
(7) |x + y|∨|x − y| = |x| + |y|.
Proof. (1) Using Theorem 1.3 we see that
and x ∧ y = 1
2 x + y −|x− y| .
x = x ∨ y − y + x ∧ y =(x − y) ∨ (y − y)+x ∧ y
= (x − y) ∨ 0+x ∧ y =(x − y) + + x ∧ y.
(2) For the first identity note that
x + y + |x − y| = x + y +(x − y) ∨ (y − x)
= (x + y)+(x − y) ∨ (x + y)+(y − x)
= (2x) ∨ (2y) =2(x ∨ y) .