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arXiv · 2610.17846

Order Boundedness of Order Continuous Orthogonally Additive Operators

Ayşe Uyar
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Abstract

We prove that, under suitable assumptions, the space of order continuous orthogonally additive operators between Riesz spaces is a Riesz space. As a consequence, we give a negative answer to an open problem posed by Mykhaylyuk and Popov concerning the order continuity of the modulus of an order continuous orthogonally additive operator. We also show that, under an appropriate condition on the domain, every order continuous orthogonally additive operator is order bounded, obtaining an orthogonally additive analogue of the corresponding result for linear operators. Moreover, we establish the order boundedness result in ZF for atomic Riesz spaces and in ZF with dependent choice (DC) for Dedekind complete Riesz spaces. Finally, we characterize when every orthogonally additive operator is an abstract Uryson operator.

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Ayşe Uyar. 2026-10-13. Order Boundedness of Order Continuous Orthogonally Additive Operators. https://arxiv.org/abs/2610.17846

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