Abstract

In this article we explore orthogonally additive (nonlinear) operators in vector lattices. First we investigate the lateral order on vector lattices and show that with every element \(e\) of a \(C\)-complete vector lattice \(E\) is associated a lateral-to-order continuous orthogonally additive projection \(\mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}\). Then we prove that for an order bounded positive \(AM\)-compact orthogonally additive operator \(S\colon E\to F\) defined on a \(C\)-complete vector lattice \(E\) and taking values in a Dedekind complete vector lattice \(F\) all elements of the order interval \([0,S]\) are \(AM\)-compact operators as well.

中文翻译：

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关于紧致正交可加算子

摘要

在本文中，我们探讨了向量格中的正交加法（非线性）算子。首先，我们探讨矢量格子的横向顺序和显示，与每一个元素\（E \）的\（C \） -complete矢量晶格\（E \）相关联的横向按订单连续正交添加剂投影\（ \mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}\)。然后我们证明，对于定义在\(C\) -完全向量格\(E\) 上并取值的阶有界正\(AM\) -紧致正交加法算子\(S\colon E\to F\)在戴德金完全向量点阵\(F\)订单区间\([0,S]\) 的所有元素也是\(AM\) -compact 运算符。