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.
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The research was supported by the Russian Foundation for Basic Research (grant no. 21-51-46006).
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(Submitted by A. B. Muravnik)
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Pliev, M. On Compact Orthogonally Additive Operators. Lobachevskii J Math 42, 989–995 (2021). https://doi.org/10.1134/S1995080221050139
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DOI: https://doi.org/10.1134/S1995080221050139