POSITIVE OPERATORS

The Order Structure
of Positive Operators
Chapter 1
A linear operator between two ordered vector spaces that carries positive elements
to positive elements is known in the literature as a positive operator.
As we have mentioned in the preface, the main theme of this book is the
study of positive operators. To obtain fruitful and useful results the domains
and the ranges of positive operators will be taken to be Riesz spaces (vector
lattices). For this reason, in order to make the material as self-sufficient as
possible, the fundamental properties of Riesz spaces are discussed as they
are needed.
Throughout this book the symbol R will denote the set of real numbers,
N will denote the set of natural numbers, Q will denote the set of rational
numbers, and Z will denote the set of integers.
1.1. Basic Properties of Positive Operators
A real vector space E is said to be an ordered vector space whenever it
is equipped with an order relation ≥ (i.e., ≥ is a reflexive, antisymmetric,
and transitive binary relation on E) that is compatible with the algebraic
structure of E in the sense that it satisfies the following two axioms:
(1) If x ≥ y, then x + z ≥ y + z holds for all z ∈ E.
(2) If x ≥ y, than αx ≥ αy holds for all α ≥ 0.
An alternative notation for x ≥ y is y ≤ x. A vector x in an ordered
vector space E is called positive whenever x ≥ 0 holds. The set of all
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