POSITIVE OPERATORS

1.1. Basic Properties of Positive Operators 7
It should be noted that the identities in (2) above show that an ordered
vector space is a Riesz space if and only if the absolute value |x| = x ∨ (−x)
exists for each vector x.
In a Riesz space, two elements x and y are said to be disjoint (in
symbols x ⊥ y) whenever |x|∧|y| = 0 holds. Note that according to part (5)
of Theorem 1.7 we have x ⊥ y if and only if |x+y| = |x−y|. Two subsets A
and B of a Riesz space are called disjoint (denoted A ⊥ B) ifa ⊥ b holds
for all a ∈ A and all b ∈ B.
If A is a nonempty subset of a Riesz space E, then its disjoint complement
A d is defined by
A d := x ∈ E : x ⊥ y for all y ∈ A .
We write A dd for (A d ) d . Note that A ∩ A d = {0}.
If A and B are subsets of a Riesz space, then we shall employ in this
book the following self-explanatory notation:
|A| := |a|: a ∈ A
A + := a + : a ∈ A
A − := a − : a ∈ A
A ∨ B := a ∨ b: a ∈ A and b ∈ B
A ∧ B := a ∧ b: a ∈ A and b ∈ B
x ∨ A := x ∨ a: a ∈ A
x ∧ A := x ∧ a: a ∈ A
The next theorem tells us that every Riesz space satisfies the infinite
distributive law.
Theorem 1.8 (The Infinite Distributive Law). Let A be a nonempty subset
of a Riesz space. If sup A exists, then for each vector x the supremum of
the set x ∧ A exists and
sup(x ∧ A) =x ∧ sup A.
Similarly, if inf A exists, then inf(x ∨ A) exists for each vector x and
inf(x ∨ A) =x ∨ inf A.
Proof. Assume that sup A exists. Let y =supA and fix some vector x.
Clearly, for each a ∈ A we have x ∧ a ≤ x ∧ y, i.e., x ∧ y is an upper bound
of the set x ∧ A. To see that x ∧ y is the least upper bound of the set
x ∧ A, assume that some vector z satisfies x ∧ a ≤ z for all a ∈ A. Sincefor
each a ∈ A we have a = x ∧ a + x ∨ a − x ≤ z + x ∨ y − x, it follows that
y ≤ z + x ∨ y − x. This implies x ∧ y = x + y − x ∨ y ≤ z, and from this we