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Order Continuity of Orthogonally Additive Operators

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Abstract

The present article aims to fill the gaps in known implications between the order continuity of an orthogonally additive operator T and its modulus |T| as well as some partial order continuities of T. We prove several results showing that, under some assumptions on Riesz spaces EF if \(T :E \rightarrow F\) is order continuous then so is |T|, however the converse is false for \(E = F = \mathbb R\). It is known that a horizontally-to-order continuous regular linear operator from a Riesz space E with the principal projection property to a Dedekind complete Riesz space F is order continuous. This is not longer true for orthogonally additive operators. We prove that, under mild assumptions, an order bounded orthogonally additive operator \(T :E \rightarrow F\) is order continuous if and only if T is uniformly-to-order continuous and the envelope operator \(\widehat{T} :E \rightarrow F\) is horizontally-to-order continuous. Some open questions remain unsolved.

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Notes

  1. component, in another terminology.

  2. disjointly continuous, in another terminology.

  3. or a Popov operator, in the terminology of [13].

  4. actually, one can show that this net strongly tends to zero.

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Fotiy, O., Krasikova, I., Pliev, M. et al. Order Continuity of Orthogonally Additive Operators. Results Math 77, 5 (2022). https://doi.org/10.1007/s00025-021-01543-x

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