POSITIVE OPERATORS

1.2. Extensions of Positive Operators 23
1.2. Extensions of Positive Operators
In this section we shall gather some basic extension theorems for operators,
and, in particular, for positive operators.
A function p: G → F , where G is a (real) vector space and F is an
ordered vector space, is called sublinear whenever
(a) p(x + y) ≤ p(x)+p(y) for all x, y ∈ G, and
(b) p(λx) =λp(x) for all x ∈ G and all λ ≥ 0.
The next result is the most general version of the classical Hahn–Banach
extension theorem. This theorem plays a fundamental role in modern analysis
and without any doubt it will be of great importance to us here. It is
due to H. Hahn [74] and S. Banach [30].
Theorem 1.25 (Hahn–Banach). Let G be a (real) vector space, F a Dedekind
complete Riesz space, and let p: G → F be a sublinear function. If H is
a vector subspace of G and S : H → F is an operator satisfying S(x) ≤ p(x)
for all x ∈ H, then there exists some operator T : G → F such that:
(1) T = S on H, i.e., T is a linear extension of S to all of G.
(2) T (x) ≤ p(x) holds for all x ∈ G.
Proof. The critical step is to show that S has a linear extension satisfying
(2) on an arbitrary vector subspace generated by H and one extra vector. If
this is done, then an application of Zorn’s lemma guarantees the existence
of an extension of S to all of G with the desired properties.
To this end, let x/∈ H, andletV = y + λx: y ∈ H and λ ∈ R . If
T : V → F is a linear extension of S, then
T (y + λx) =S(y)+λT (x)
must hold true for all y ∈ H and all λ ∈ R. Putz = T (x). To complete the
proof, we must establish the existence of some z ∈ F such that
S(y)+λz ≤ p(y + λx) (⋆)
holds for all y ∈ H and λ ∈ R. Forλ>0, (⋆) is equivalent to
S(y)+z ≤ p(y + x)
for all y ∈ H, while for λ