Abstract
In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map \(T:{\mathcal {I}}\rightarrow W\) defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space \(E\times F\). Then we prove that a positive orthogonally bi-additive operator \(T:E\times F\rightarrow W\) is laterally-to-order continuous if and only if the kernel of each S with \(0\le S\le T\) is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator \(T:E\times F\rightarrow W\).
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Acknowledgements
Nonna Dzhusoeva was supported by the Ministry of Science and Education of Russian Federation (grant number \(075-02-2023-914\)).
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Dzhusoeva, N., Mazloeva, M. Orthogonally Bi-additive Operators-II. Results Math 78, 182 (2023). https://doi.org/10.1007/s00025-023-01957-9
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DOI: https://doi.org/10.1007/s00025-023-01957-9
Keywords
- Orthogonally bi-additive operator
- regular operator
- positive operator
- laterally-to-order continuous operator
- vector lattice