On the Calculus of Order Bounded Operators

Abstract

Abstract. This article deals with the Abramovich calculus of order bounded operators. Key words: fragment, Freudenthal property, order bounded operator, Riesz–Kantorovichtheorem 1. Heuristic Preliminaries1.1. We start with a few well-known definitions and facts. Let E and F bevector lattices. The set of order bounded linear operators from E to F isdenoted by L ∼ (E,F) . The order on L (E,F) is introduced by the positivecone L + (E,F) comprising all T ∈ L ∼ (E,F) such that T(E + ) ⊂ F + . Through-out E + stands for the positive cone of E . The celebrated Riesz–Kantorovichtheorem asserts that if F is a Dedekind complete vector lattice then so is L ∼ (E,F) . Moreover, several convenient formulas are available for calculat-ing the lattice operations in L ∼ (E,F) . For example, given x ∈ E + and S,T ∈ L ∼ (E,F) we see (S ∨ T)x = sup{ Sy + Tz : y,z ∈ E + ; x = y + z }; (1) (S ∧ T)x = inf{ Sy + Tz : y,z ∈ E + ; x = x 1 + x 2 } . (2)Analogous explicit expressions exist for finite and infinite lattice oper-ations as well as for the modulus, positive and negative parts of anorder bounded operator. There are bulkier versions of these formulas withsuprema over upward directed sets and infima over downward directed setswhich are sometimes essential in applications. The collection of these andall similar formulas is usually referred to as the