The Rosenthal theorem on the decomposition for operators in L 1 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 L 1 is a projective component, which yields the known fact that a sum of narrow operators in L 1 is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L 1 , in particular the Liu decomposition.
Preliminary InformationWe use the standard terminology of the theories of classical Banach spaces [1, 2], vector lattices, and positive operators [3]. By L ( X , Y ) we denote the set of all linear bounded operators that act from a Banach space X into a Banach space Y. We use the symbol L ( X ) as an abbreviation for L ( X , X ). The word "subspace" in a Banach space means a closed subspace.We denote the set of all Lebesgue-measurable subsets of [0, 1] by Σ , the Lebesgue measure on Σ by μ, and the characteristic function of the set A by χ ( A ). In addition, we use the following abbreviations:Below, we formulate the Kalton theorem on the representation of linear bounded operators that act from L 1 into L 1 [4]. Theorem 1 (Kalton representation theorem). For any T ∈ L ( L 1 ), there exists a weak*-measurable function μ t from the interval [ 0, 1 ] into the set M [ 0, 1 ] of all Borel regular measures on [ 0, 1 ] such that, for any x ∈ L 1 , one has T x ( t ) = x d t ( ) ( ) τ μ τ ∫ (1) almost everywhere. Conversely, every weak*-measurable function μ t : [ 0, 1 ] → M [ 0, 1 ] defines some operator T ∈ L ( L 1 ) according to (1). Representing the measure μ t as the sum of the atomic part μ t a = a t n t n n ( ) ( ) δ σ = ∞ ∑ 1 (where δ τ is the Dirac measure) and the continuous (i.e., atomless) part μ t c , we obtain the following representation of the operator T (see [5]):
