Elementary theory

Ideas connected with positivity have permeated a good deal of the work in this book. For operators defined on a normed lattice, it is natural to consider a “summing” norm that is defined in a way that pays attention to the order structure. The simplest way to do this is to restrict to positive elements in the definition of φ1. The resulting “cone-summing” norm gives rise to a theory that parallels closely (and in places more simply) the most successful parts of the theory of 1-summing and 2-summing norms. It also provides a proper setting for our sporadic earlier remarks on positive operators. The concept and the basic results are due to Schlotterbeck (1971).

To set the scene, we need a few very elementary concepts and results relating to normed lattices. The definition was given in Section 0. The set {x: x ≥ 0) in a linear lattice X is called the positive cone, and will be denoted by X+. The supremum of the two elements x, y is denoted by x ∨ y, the infimum by x ∧ y.