POSITIVE OPERATORS

Preface
Linear operators have been studied in various contexts and settings in the
past. Their study is a subject of great importance both to mathematics
and to its applications. The present book deals mainly with the special
class of linear operators known as positive operators. A linear operator
between two ordered vector spaces that carries positive elements to positive
elements is referred to as a positive operator. For instance, the linear
operator T : C[0, 1] → C[0, 1], defined by
x
Tf(x) = f(t) dt ,
carries positive functions of C[0, 1] to positive functions, and is thus an example
of a positive operator. The material covered in the book has some
overlap with the material in the books by H. H. Schaefer [174] and A. C. Zaanen
[197]. However, our attention is mostly focused on recent developments
of the subject and the overlap with the above-mentioned books is kept to
a minimum. On the other hand, we have intentionally covered those topics
where the ingredient of positivity allowed us to obtain beautiful and elegant
results.
It is well known that many linear operators between Banach spaces arising
in classical analysis are in fact positive operators. For this reason, in this
book positive operators are studied in the setting of Riesz spaces and Banach
lattices. In order to make the book as self-sufficient as possible, some basic
results from the theory of Riesz spaces and Banach lattices are included with
proofs as needed. On the other hand, we assume that the reader is familiar
with the elementary concepts of real analysis and functional analysis.
0
xiii