Factorization theorems of Arendt type for additive monotone mappings

Abstract

We deal with additive monotone mappings defined on a lattice-ordered Abelian group and having values in a Dedekind complete Riesz space and which are invariant with respect to some representation of an amenable semigroup. Using a Hahn–Banach-type theorem of Zbigniew Gajda, we obtain generalizations of factorization theorems obtained in 1984 by Wolfgang Arendt for positive linear operators. The theorems of Arendt are generalized in two directions. First, we extend these results from the case of linear operators acting between Riesz spaces to the case of additive mappings between lattice-ordered Abelian groups. Second, we study mappings which are invariant with respect to a semigroup representation.As an application of the results obtained, we show some property of composition operators between spaces of additive functions acting between lattice-ordered groups.