1683-3414 1814-0807 Владикавказский математический журнал Vladikavkaz Mathematical Journal Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН) On Band Preserving Operators on Complex Vector Lattices О нерасширяющих операторах в комплексных векторных решетках 10.46698/h7168-4322-6544-h 18570 03 2026 28 1 7 15 https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=18599&SECTION_ID=658 Гутнова А. К. Gutnova A. K. gutnovaalina@gmail.com Абасов М. Н. Abasov N. abasovn@mail.ru Северо-Осетинский государственный университет им. К.Л. Хетагурова, РОССИЯ, 362025, Владикавказ, ул. Ватутина, 44-46 North Ossetian State University, 44-46 Vatutin Street, Vladikavkaz 362025, Russia Московский государственный технический университет им. Н. Э. Баумана, Россия, Москва, 2-я Бауманская улица, 5, стр. 4 Bauman Moscow State Technical University, 5, Bldg. 4 2nd Baumanskaya St., Moscow 105005, Russia 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 \(\{\mathcal{T}_{I,I}\}^{\perp\perp}\). Данная заметка продолжает цикл исследований, инициированных работой [1]. В статье рассматривается подкласс так называемых "нерасширяющих" ортогонально аддитивных операторов, заданных на комплексификации \(E_{\mathbb{C}}\) равномерно полной векторной решетки и принимающих значения в \(E\). Будем говорить, что ортогонально аддитивный оператор \(\mathcal{T}\colon E_{\mathbb{C}}\to E\) является нерасширяющим, если \(\mathcal{T}(w)\in \{|w|\}^{\perp\perp}\) для каждого элемента \(w\) из \(E_{\mathbb{C}}\). Вводится и изучается класс элементарных нерасширяющих операторов, которые представляют собой комплексные расширения \(\mathcal{T}_{T,S}\), построенные из пар вещественных операторов \(T, S\colon E \to E\), коммутирующих со всеми нерасширяющими проекторами. Показано, что такие операторы не только являются нерасширяющими, но и регулярны. Представлено несколько примеров таких операторов и установлено, что действительное векторное пространство \(\mathcal{N}(E_{\mathbb{C}}, E)\) всех элементарных нерасширяющих ортогонально аддитивных операторов является подрешеткой \(\mathcal{OA}_r(E_{\mathbb{C}}, E)\) - порядково полной векторной решетки всех регулярных ортогонально аддитивных операторов из \(E_{\mathbb{C}}\) в \(E\). Показано, что операции решетки в этой подрешетке вычисляются поточечно, отражая структуру пространства \(E\), с явными формулами для супремума, инфимума, положительной части, отрицательной части и модуля. Кроме того, установлено, что \(\mathcal{N}(E_{\mathbb{C}}, E)\) содержится в полосе, порожденной комплексным расширением единичного оператора \(\{\mathcal{T}_{I,I}\}^{\perp\perp}\). ортогонально аддитивный оператор нерасширающий оператор регулярный оператор порядковый проектор векторная решетка комплексификация orthogonally additive operator band preserving operator regular operator order projection vector lattice complexification. The research was supported by the Ministry of Science and High Education, agreement no. 075-02-2026-1324. 1 Pliev, M. and Sukochev, F. Orthogonally Additive Operators on Complex Vector Lattices,Journal of Mathematical Analysis and Applications, 2025, vol. 541, no. 2, article no. 128719.DOI: 10.1016/ j.jmaa.2024.128719. Pliev, M. and Sukochev, F. Orthogonally Additive Operators on Complex Vector Lattices, Journal of Mathematical Analysis and Applications, 2025, vol. 541, no. 2, article no. 128719. DOI: 10.1016/ j.jmaa.2024.128719. 2 Boulabiar, K. Recent Trends on Order Bounded Disjointness Preserving Operators,Irish Mathematical Society Bulletin, 2008, vol. 62, pp. 43-69.DOI: 10.33232/BIMS.0062.43.69. Boulabiar, K. Recent Trends on Order Bounded Disjointness Preserving Operators, Irish Mathematical Society Bulletin, 2008, vol. 62, pp. 43-69. DOI: 10.33232/BIMS.0062.43.69. 3 Mazon, J. M. and Segura de Leon, S. Order Bounded Orthogonally Additive Operators,Romanian Journal of Pure and Applied Mathematics, 1990, vol. 35, no. 4, pp. 329-353. Mazon, J. M. and Segura de Leon, S. Order Bounded Orthogonally Additive Operators, Romanian Journal of Pure and Applied Mathematics, 1990, vol. 35, no. 4, pp. 329-353. 4 Erkursun Ozcan, N. and Pliev, M. On Orthogonally Additive Operators in \(C\)-Complete Vector Lattices,Banach Journal of Mathematical Analysis, 2022, vol. 16, article no. 6. DOI: 10.1007/s43037-021-00158-2. Erkursun Ozcan, N. and Pliev, M. On Orthogonally Additive Operators in \(C\)-Complete Vector Lattices, Banach Journal of Mathematical Analysis, 2022, vol. 16, article no. 6. DOI: 10.1007/s43037-021-00158-2. 5 Feldman, W. A. A Factorization for Orthogonally AdditiveOperators on Banach Lattices, Journal of Mathematical Analysis and Applications,2019, vol. 472, no. 1, pp. 238-245. DOI: 10.1016/ j.jmaa.2018.11.021. Feldman, W. A. A Factorization for Orthogonally Additive Operators on Banach Lattices, Journal of Mathematical Analysis and Applications, 2019, vol. 472, no. 1, pp. 238-245. DOI: 10.1016/ j.jmaa.2018.11.021. 6 Fotiy, O., Kadets, V. and Popov, M. Some Remarks on Orthogonally Additive Operators,Positivity, 2023, vol. 27, article no. 57. DOI: 10.1007/s11117-023-01008-1. Fotiy, O., Kadets, V. and Popov, M. Some Remarks on Orthogonally Additive Operators, Positivity, 2023, vol. 27, article no. 57. DOI: 10.1007/s11117-023-01008-1. 7 Mykhaylyuk, V. and Popov, M. \(\varepsilon\)-Shading Operator on Riesz Spacesand Order Continuity of Orthogonally Additive Operators,Results in Mathematics, 2022, vol. 77, article no. 209.DOI: 10.1007/s00025-022-01742-0. Mykhaylyuk, V. and Popov, M. \(\varepsilon\)-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators, Results in Mathematics, 2022, vol. 77, article no. 209. DOI: 10.1007/s00025-022-01742-0. 8 Popov, M. Banach Lattices of Orthogonally Additive Operators,Journal of Mathematical Analysis and Applications, 2022, vol. 514, no. 1, article no. 126279.DOI: 10.1016/j.jmaa.2022.126279. Popov, M. Banach Lattices of Orthogonally Additive Operators, Journal of Mathematical Analysis and Applications, 2022, vol. 514, no. 1, article no. 126279. DOI: 10.1016/j.jmaa.2022.126279. 9 Tulu, D. and Turan, B. Extension and Restriction of Orthogonally Additive Operators,Siberian Mathematical Journal, 2025, vol. 66, pp. 199-206.DOI: 10.1134/S003744662501015X. Tulu, D. and Turan, B. Extension and Restriction of Orthogonally Additive Operators, Siberian Mathematical Journal, 2025, vol. 66, pp. 199-206. DOI: 10.1134/S003744662501015X. 10 Abasov, N. M. On Band Preserving Orthogonally Additive Operators,Siberian Electronic Mathematical Reports, 2021, vol. 18, no. 1, pp. 495-510.DOI: 10.33048/semi.2021.18.036. Abasov, N. M. On Band Preserving Orthogonally Additive Operators, Siberian Electronic Mathematical Reports, 2021, vol. 18, no. 1, pp. 495-510. DOI: 10.33048/semi.2021.18.036. 11 Abasov, N. On a Band Generated by a DisjointnessPreserving Orthogonally Additive Operator, Lobachevskii Journal of Mathematics,2021, vol. 42, no. 5, pp. 851-856. DOI: 10.1134/ S1995080221050024. Abasov, N. On a Band Generated by a Disjointness Preserving Orthogonally Additive Operator, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 5, pp. 851-856. DOI: 10.1134/ S1995080221050024. 12 Abasov, N. M., Dzhusoeva, N. A. and Pliev, M. A. Diffuse Orthogonally Additive Operators,Sbornik: Mathematics, 2024, vol. 215, no. 1, pp. 1-27. DOI: 10.4213/sm9909e. Abasov, N. M., Dzhusoeva, N. A. and Pliev, M. A. Diffuse Orthogonally Additive Operators, Sbornik: Mathematics, 2024, vol. 215, no. 1, pp. 1-27. DOI: 10.4213/sm9909e. 13 Abasov, N. and Pliev, M. Disjointness Preserving OrthogonallyAdditive Operators in Vector Lattices, Banach Journal of Mathematical Analysis,2018, vol. 12, no. 3, pp. 730-750. DOI: 10.1215/17358787-2018-0001. Abasov, N. and Pliev, M. Disjointness Preserving Orthogonally Additive Operators in Vector Lattices, Banach Journal of Mathematical Analysis, 2018, vol. 12, no. 3, pp. 730-750. DOI: 10.1215/17358787-2018-0001. 14 Turan, B. and Tulu, D. On Orthogonally Additive Band Operators andOrthogonally Additive Disjointness Preserving Operators,Turkish Journal of Mathematics, 2023, vol. 47, no. 4, article no. 15.DOI: 10.55730/1300-0098.3425. Turan, B. and Tulu, D. On Orthogonally Additive Band Operators and Orthogonally Additive Disjointness Preserving Operators, Turkish Journal of Mathematics, 2023, vol. 47, no. 4, article no. 15. DOI: 10.55730/1300-0098.3425. 15 Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Dordrecht, Springer, 2006. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Dordrecht, Springer, 2006. 16 Kusraev, A. G. Dominated Operators, Kluwer Academic Publishers, 2000. Kusraev, A. G. Dominated Operators, Kluwer Academic Publishers, 2000. 17 Vulikh, B. Z. Introduction to the Theory of Partially Ordered Spaces,Gronngen, Wolter-Noordhoff Scientific Publications, LTD, 1967. Vulikh, B. Z. Introduction to the Theory of Partially Ordered Spaces, Gronngen, Wolter-Noordhoff Scientific Publications, LTD, 1967. 18 Mykhaylyuk, V., Pliev, M. and Popov, M. The Lateral Order on Riesz Spacesand Orthogonally Additive Operators, Positivity, 2021, vol. 25, pp. 291-327.DOI: 10.1007/s11117-020-00761-x. Mykhaylyuk, V., Pliev, M. and Popov, M. The Lateral Order on Riesz Spaces and Orthogonally Additive Operators, Positivity, 2021, vol. 25, pp. 291-327. DOI: 10.1007/s11117-020-00761-x. 19 Meyer-Nieberg, P. Banach Lattices, Berlin, Springer, 1991. Meyer-Nieberg, P. Banach Lattices, Berlin, Springer, 1991. 20 Dzhusoeva, N., Huang, J., Pliev, M. and Sukochev, F. Lateral Order on Complex Vector Lattices and Narrow Operators,Mathematische Nachrichten, 2023, vol. 296, no. 11, pp. 5157-5170. DOI: 10.1002/ mana.202200415. Dzhusoeva, N., Huang, J., Pliev, M. and Sukochev, F. Lateral Order on Complex Vector Lattices and Narrow Operators, Mathematische Nachrichten, 2023, vol. 296, no. 11, pp. 5157-5170. DOI: 10.1002/ mana.202200415. 21 Appell, J. and Zabrejko, P. P. Nonlinear Superposition Operators,Cambridge, Cambridge University Press, 1990. Appell, J. and Zabrejko, P. P. Nonlinear Superposition Operators,Cambridge, Cambridge University Press, 1990.