POSITIVE OPERATORS

Historical Foreword
′ Oλβιos óστιs ιστoρ´ιηs ´ɛσχɛ µ´αθησιν .
Eυριπ´ιδηs
Wise is the person who knows history.
Euripides
Positive operators made their debut at the beginning of the nineteenth century.
They were tied to integral operators (whose study triggered the birth of functional
analysis) and to matrices with nonnegative entries. However, positive operators
were investigated in a systematic manner much later. Their study followed closely
the development of Riesz spaces. An address of F. Riesz in 1928 On the decomposition
of linear functionals [166] (into their positive and negative parts), at the
International Congress of Mathematicians in Bologna, Italy, marked the beginnings
of the study of Riesz spaces and positive operators. The theory of Riesz spaces was
developed axiomatically in the mid-1930s by H. Freudenthal [67] and L. V. Kantorovich
[85, 86, 87, 88, 89, 90, 91]. Positive operators were also introduced and
studied in the mid-1930s by L. V. Kantorovich, and they made their first textbook
appearance in the 1940 edition of G. Birkhoff’s book Lattice Theory [37]. Undoubtedly,
the systematic study of positive operators was originated in the 1930s
by F. Riesz, L. V. Kantorovich and G. Birkhoff.
In the 1940s and early 1950s one finds very few papers on positive operators.
In this period, the main contributions came from the Soviet school (L. V. Kantorovich,
M. G. Krein, A. G. Pinsker, M. A. Rutman, B. Z. Vulikh) and the Japanese
school (H. Nakano, K. Yosida, T. Ogasawara, and their students). In the 1950 the
book Functional Analysis in Partially Ordered Spaces [92] by L. V. Kantorovich,
B. Z. Vulikh, and A. G. Pinsker appeared in the Soviet literature. This book (that
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