Chapter 2 Positive operators

Abstract

This chapter discusses the theory of positive operators that is a distinguished and significant part of the field of general operator theory. The extra feature of this part of operator theory is the existence of an order on the spaces involved. The two main objectives of the chapter are to discuss the relationships between the general operators and the positive operators and to demonstrate the effects the order structure has on the general operators, acting between the Banach lattices. Only Archimedean vector lattices are discussed in the chapter. A normed vector lattice is a vector lattice equipped with a lattice norm. A norm complete normed vector lattice is called a Banach lattice. A Banach lattice with an order continuous Levi norm is usually referred to as a Kantorovich-Banach space or as a KB-space.