Factoring Compact Operators and Approximable Operators

Abstract

Our paper is concerned with two topics. The first one is represented by aversion of Figiel’s and Johnson’s theorem on the factorization of compact operators adapted to the framework of ordered Banach spaces. Namely, we prove that every compact operator from a Banach space to an ordered Banach space with closed generating cone (respectively, a Banaöh lattice) factors, with compact factors, through a reflexive lattice-ordered Banach space with closed generating cone, the second factor being positive (respectively, a reflexive lattice-ordered Banach space with continuous modulus, the second factcr being a Riesz homomorphism). The second topic is provided by a discussion of the factorization of approximable operators between :Banach lattices. We prove that every such operator $U$ factors through a reflexive Banach lattice with an unconditional basis, the factors being compact and one of them being positive. We also give a necessary and sufficient condition on $U$ under which both factors in the mentioned factorization can be taken to be differences of positive compact operators.