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.

中文翻译：

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拟巴拿赫格中带值的正序连续算子的最优扩展

本文的目标是提出一些在 Dedekind 完全拟巴拿赫格中具有值的拟巴拿赫函数空间上的正序连续和 $\sigma $ -阶连续算子的最优扩展方法。这种算子的最优扩展是 Bartle-Dunford-Schwartz 型积分的最小扩展。还表明，如果一个正算子将阶收敛序列发送到拟同模收敛序列，那么它的最优扩展就是 Bartle-Dunford-Schwartz 型积分。