Optimal extension of positive order continuous operators with values in quasi-banach lattices

The goal of this article is to present some method of optimal extension of positive order continuous and $\sigma$-order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order convergent sequences to quasinorm convergent sequences, then its optimal extension is the Bartle–Dunford–Schwartz type integral.