Lattices of Operators

Abstract

The present chapter is primarily concerned with vector lattices of linear operators between Banach lattices or, more precisely, with the problem of exhibiting significant classes of linear operators possessing a (linear) modulus. Up to the late 1960’s, the available knowledge in this area appeared somewhat fragmentary and incoherent; above all, however, of little relevance to the mainstream of operator theory. The fact is that order bounded operators are very intimately related to operators of recognized significance, as is evidenced by Proposition 5.7 (iii); more specifically, a linear map from an AM-space into an AL-space is order bounded (equivalently, possesses a modulus) if and only if it is integral in the sense of Grothendieck [1955a]. Combined with a judicious use of AM- and AL-spaces as structural companions of general Banach lattices (cf. § 3), the systematic exploitation of those facts permits many interesting applications: For example, a fairly easy access to the theory of integral and absolutely summing maps, construction of Banach lattice tensor products with identification of their duals, clarification of the relations between kernel and nuclear operators, and others.