POSITIVE OPERATORS

1.3. Order Projections 31
Exercises
1. Let E and F be two Riesz spaces with F Dedekind complete, and let A
be an ideal of E. ForeachT ∈Lb(E,F) letR(T ) denote the restriction
of T to A. Show that the positive operator R: Lb(E,F) →Lb(A, F )
satisfies
R(S ∨ T )=R(S) ∨R(T ) and R(S ∧ T )=R(S) ∧R(T )
for all S, T ∈Lb(E,F). 1
2. For two arbitrary solid sets A and B of a Riesz space show that:
(a) A + B is a solid set.
(b) If 0 ≤ c ∈ A + B holds, then there exist 0 ≤ a ∈ A and 0 ≤ b ∈ B
with c = a + b.
3. Let T : E → F be a positive operator between two Riesz spaces with F
Dedekind complete. If two ideals A and B of E satisfy A ⊥ B, then show
that:
(a) TA ∧ TB =0.
(b) The ideal A + B satisfies TA+B = TA + TB = TA ∨ TB.
4. As usual, ℓ∞ denotes the Riesz space of all bounded real sequence, and
c the Riesz subspace of ℓ∞ consisting of all convergent sequences. If
φ: c → R is the positive operator defined by
φ(x1,x2,...) = lim
n→∞ xn ,
then show that φ has a positive linear extension to all of ℓ∞.
1.3. Order Projections
In this section we shall study a special class of positive operators known as
order (or band) projections. Before starting our discussion, let us review a
few properties of order dense Riesz subspaces. Recall that a Riesz subspace
G of a Riesz space E is said to be order dense in E whenever for each
0