POSITIVE OPERATORS

1.1. Basic Properties of Positive Operators 9
of functional analysis (notably the function spaces and Lp-spaces) are Archimedean.
For this reason, the focus of our work will be on the study of
positive operators between Archimedean Riesz spaces. Accordingly:
• Unless otherwise stated, throughout this book all Riesz spaces will
be assumed to be Archimedean.
The starting point in the theory of positive operators is a fundamental
extension theorem of L. V. Kantorovich [91]. The importance of the result
lies in the fact that in order for a mapping T : E + → F + to be the restriction
of a (unique) positive operator from E to F it is necessary and sufficient to
be additive on E + . The details follow.
Theorem 1.10 (Kantorovich). Suppose that E and F are two Riesz spaces
with F Archimedean. Assume also that T : E + → F + is an additive mapping,
that is, T (x + y) =T (x) +T (y) holds for all x, y ∈ E + . Then T
has a unique extension to a positive operator from E to F . Moreover, the
extension (denoted by T again) is given by
for all x ∈ E.
T (x) =T (x + ) − T (x − )
Proof. Let T : E + → F + be an additive mapping. Consider the mapping
S : E → F defined by
S(x) =T (x + ) − T (x − ) .
Clearly, S(x) =T (x) foreachx∈E + . So, the mapping S extends T to
all of E. Since x = x + − x− for each x ∈ E, it follows that S is the only
possible linear extension of T to all of E. Therefore, in order to complete
the proof, we must show that S is linear. That is, we must prove that S is
additive and homogeneous.
For the additivity of S start by observing that if any vector x ∈ E
can be written as a difference of two positive vectors, say x = x1 − x2
with x1,x2 ∈ E + , then S(x) =T (x1) − T (x2) holds. To see this, fix any
x ∈ E and assume that x = x + − x− = x1 − x2, where x1,x2 ∈ E + . Then
x + + x2 = x1 + x− , and so the additivity of T on E + yields
T (x + )+T (x2) =T (x + + x2) =T (x1 + x − )=T (x1)+T (x − )