Cone isomorphisms and almost surjective operators

Let EE be a Banach lattice and FF a Banach space. A bounded linear operator T:E→FT:E→F is an isomorphism on the positive cone of EE if and only if T∗T∗ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→ET:F→E is almost surjective if and only if T∗T∗ is an isomorphism on the positive cone of F∗F∗.