Abstract: In this article we continue an investigation of orthogonally additive operators on complex vector lattices started in [1]. We study the special class of so called band preserving orthogonally additive operators defined on the complexification \(E_{\mathbb{C}}\) of a uniformly complete vector lattice \(E\) and taking values in \(E\). We say that an orthogonally additive operator \(\mathcal{T}\colon E_{\mathbb{C}}\to E\) is band preserving if \(\mathcal{T}(w)\in \{|w|\}^{\perp\perp}\) for every element \(w\) of \(E_{\mathbb{C}}\). The authors introduce and study the class of elementary band preserving operators, which are complex extensions \(\mathcal{T}_{T,S}\) constructed from pairs of real operators \(T, S \colon E \to E\) that commute with all band projections. It is demonstrated that such operators are not only band preserving, but also regular. A central result of the work is that the set \(\mathcal{N}(E_{\mathbb{C}}, E)\) of all elementary band preserving operators constitutes a vector sublattice within the Dedekind complete vector lattice \(\mathcal{OA}_r(E_{\mathbb{C}}, E)\) of all regular orthogonally additive operators. The lattice operations in this sublattice are shown to be calculated pointwise, mirroring the structure of the target space \(E\), with explicit formulas provided for the supremum, infimum, positive part, negative part, and modulus. Furthermore, it is established that \(\mathcal{N}(E_{\mathbb{C}}, E)\) is contained within the band generated by the complex extension of the identity operator \(\{\mathcal{T}_{I,I}\}^{\perp\perp}\).
Keywords: orthogonally additive operator, band preserving operator, regular operator, order projection, vector lattice, complexification.
Fund name: The research was supported by the Ministry of Science and High Education, agreement no. 075-02-2026-1324.
For citation: Abasov, N. and Gutnova, A. On Band Preserving Operators on Complex Vector Lattices, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 7-15. DOI 10.46698/h7168-4322-6544-h
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