发表机构
Retired Professor of Mathematics, Istanbul, Türkiye（土耳其伊斯坦布尔退休数学教授）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个统一框架，研究Riesz空间间基于广义序收敛的$\mathcal{FG}$-序连续算子，证明其有序有界性、模保持性、序连续性与带刻画，并给出正算子的扩张定理。
AI 中文摘要
我们基于Tantrawan的广义（无界）序收敛，引入并研究了Riesz空间之间的$\mathcal{FG}$-序连续算子。该框架包含了几个熟悉的算子类，并允许在统一的环境中研究它们的性质。我们建立了有序有界性和格论结果；特别地，我们证明了强序连续算子自动是有序有界的，并证明了在适当假设下，一个有序有界的$\mathcal{FG}$-序连续算子的模仍然是$\mathcal{FG}$-序连续的。后一个结果对先前提出的关于强序连续算子的一个问题给出了肯定回答。我们还建立了一个确保经典序连续的一般结果，由此可得出几个熟悉算子类的序连续结果。此外，在适当假设下，我们刻画了有序有界$\mathcal{FG}$-序连续算子的空间何时是一个带。最后，我们证明了正$\mathcal{FG}$-序连续算子的一个扩张定理，在该框架中提供了Veksler扩张定理的对应物。
英文摘要
We introduce and study $\mathcal{FG}$-order continuous operators between Riesz spaces, based on Tantrawan's generalized (unbounded) order convergence. This framework includes several familiar classes of operators and allows their properties to be studied in a unified setting. We establish order boundedness and lattice-theoretic results; in particular, we show that strongly order continuous operators are automatically order bounded and prove that the modulus of an order bounded $\mathcal{FG}$-order continuous operator remains $\mathcal{FG}$-order continuous under suitable assumptions. The latter result gives an affirmative answer to a previously posed problem for strongly order continuous operators. We also establish a general result ensuring classical order continuity, from which order continuity results for several familiar classes of operators follow. Furthermore, we characterize, under suitable hypotheses, when the space of order bounded $\mathcal{FG}$-order continuous operators is a band. Finally, we prove an extension theorem for positive $\mathcal{FG}$-order continuous operators, providing a counterpart of Veksler's extension theorem in this framework.
Comments24 pages
