Factoring compact and weakly compact operators through reflexive Banach lattices

When does a weakly compact operator between two Banach spaces factor through a reflexive Banach lattice? This paper provides some answers to this question. One of the main results: If an operator between two Banach spaces factors through a Banach lattice with weakly compact factors, then it also factors through a reflexive Banach lattice. In particular, the square of a weakly compact operator on a Banach lattice factors through a reflexive Banach lattice. Similar results hold for compact operators. For instance, the square of a compact operator on a Banach lattice factors with compact factors through a reflexive Banach lattice.