Theorems on decomposition of operators in <i>L</i>₁ and their generalization to vector lattices

Abstract

The Rosenthal theorem on the decomposition for operators in L₁ is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L₁ is a projective component, which yields the known fact that a sum of narrow operators in L₁ is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L₁, in particular the Liu decomposition.