Order continuous extensions of positive compact operators on Banach lattices

Let EE and FF be Banach lattices. Let GG be a vector sublattice of EE and T:G→FT:G→F be an order continuous positive compact (resp. weakly compact) operator. We show that if GG is an ideal or an order dense sublattice of EE, then TT has a norm preserving compact (resp. weakly compact) positive extension to EE which is likewise order continuous on EE. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of EE extends uniquely to a compact positive orthomorphism on EE.