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Narrow and \(C\)-compact orthogonally additive operators in lattice-normed spaces. (English) Zbl 07112092

Summary: In this article we consider orthogonally additive operators on lattice-normed spaces. In the first part of the article we present some examples of narrow, laterally-to-norm continuous and \(C\)-compact operators defined on a lattice-normed space and taking value in a Banach space. We show that any laterally-to-norm continuous narrow orthogonally additive operator defined on a decomposable lattice-normed space \((V, E)\) over an atomic vector lattice \(E\) with the projection property is equal to zero. In the second part we prove that the sum of two orthogonally additive operators \(T+S\) defined on a order complete, decomposable lattice-normed space \(V\) and taking value in Banach space \(X\), where \(T:V\rightarrow X\) is a laterally-to-norm continuous \(C\)-compact operator and \(S:V\rightarrow X\) is a narrow operator, is a narrow operator as well.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H99 Nonlinear operators and their properties
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[1] Abasov, N.; Megahed, AM; Pliev, M., Dominated operators from lattice-normed spaces to sequence Banach lattices, Ann. Funct. Anal., 7, 646-655 (2016) · Zbl 1365.46018 · doi:10.1215/20088752-3660990
[2] Abasov, N.; Pliev, M., On extensions of some nonlinear maps in vector lattices, J. Math. Anal. Appl., 455, 516-527 (2017) · Zbl 1459.47020 · doi:10.1016/j.jmaa.2017.05.063
[3] Abasov, N.; Pliev, M., Dominated orthogonally additive operators in lattice-normed spaces, Adv. Oper. Theory, 4, 251-264 (2019) · Zbl 06946453 · doi:10.15352/aot.1804-1354
[4] Abasov, N.; Pliev, M., Two definitions of narrow operators on Köthe-Bochner spaces, Arch. Math., 111, 167-176 (2018) · Zbl 1407.46031 · doi:10.1007/s00013-018-1172-2
[5] Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006) · Zbl 1098.47001 · doi:10.1007/978-1-4020-5008-4
[6] Aydin, A., Emelyanov, E.Yu., Erkursun Ozcan, N., Marabeh, M.A.A.: Compact-like operators in lattice-normed spaces. Indag. Math. 29(2), 633-656 (2018) · Zbl 1440.47032
[7] Buskes, G.; Dever, J., The complexification of a lattice seminormed space, Indag. Math., 25, 170-185 (2014) · Zbl 1311.46063 · doi:10.1016/j.indag.2013.05.001
[8] Diestel, J., Uhl, J.J.: Vector Measures. AMS, Providence (1977) · Zbl 0369.46039 · doi:10.1090/surv/015
[9] Feldman, WA, Lattice preserving maps on lattices of continuous functions, J. Math. Anal. Appl., 404, 310-316 (2013) · Zbl 1304.54062 · doi:10.1016/j.jmaa.2013.03.017
[10] Feldman, WA, A characterization of non-linear maps satisfying orthogonality properties, Positivity, 21, 85-97 (2017) · Zbl 06731874 · doi:10.1007/s11117-016-0408-2
[11] Feldman, WA, A factorization for orthogonally additive operators on Banach lattices, J. Math. Anal. Appl., 472, 238-245 (2019) · Zbl 1419.46016 · doi:10.1016/j.jmaa.2018.11.021
[12] Humenchuk, HI, On the sum of narrow and finite-rank orthogonally additive operator, Ukr. Math. J., 67, 1831-1837 (2016) · Zbl 1482.47109 · doi:10.1007/s11253-016-1193-6
[13] Kantorovich, LV, On a class of functional equations, Dokl. Akad. Nauk SSSR, 4, 211-216 (1936)
[14] Kusraev, A.G.: Dominated Operators. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0983.47025 · doi:10.1007/978-94-015-9349-6
[15] Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. 1. North Holland Publ. Comp, Amsterdam (1971) · Zbl 0231.46014
[16] Maslyuchenko, O.; Mykhaylyuk, V.; Popov, M., A lattice approach to narrow operators, Positivity, 13, 459-495 (2009) · Zbl 1183.47033 · doi:10.1007/s11117-008-2193-z
[17] Mazón, JM; Segura de León, S., Uryson operators, Rev. Roum. Math. Pures Appl., 35, 431-449 (1990) · Zbl 0717.47031
[18] Mykhaylyuk, V.; Popov, M., On sums of narrow operators on Köthe function space, J. Math. Anal. Appl., 404, 554-561 (2013) · Zbl 1316.47034 · doi:10.1016/j.jmaa.2013.03.008
[19] Mykhaylyuk, V., On the sum of narrow and a compact operators, J. Funct. Anal., 266, 5912-5920 (2014) · Zbl 1306.47028 · doi:10.1016/j.jfa.2014.01.012
[20] Orlov, V.; Pliev, M.; Rode, D., Domination problem for \(AM\)-compact abstract Uryson operators, Arch. Math., 107, 543-552 (2016) · Zbl 06668331 · doi:10.1007/s00013-016-0937-8
[21] Plichko, A.; Popov, M., Symmetric function spaces on atomless probability spaces, Diss. Math. (Rozpr. Mat.), 306, 1-85 (1990) · Zbl 0715.46011
[22] Pliev, M.; Popov, M., Narrow orthogonally additive operators, Positivity, 18, 641-667 (2014) · Zbl 1331.47078 · doi:10.1007/s11117-013-0268-y
[23] Pliev, M., Narrow operators on lattice-normed spaces, Open Math., 9, 1276-1287 (2011) · Zbl 1253.47024
[24] Pliev, M.; Fang, X., Narrow orthogonally additive operators in lattice-normed spaces, Sib. Math. J., 58, 134-141 (2017) · Zbl 1377.47021 · doi:10.1134/S0037446617010177
[25] Pliev, M., Domination problem for narrow orthogonally additive operators, Positivity, 21, 23-33 (2017) · Zbl 1420.47017 · doi:10.1007/s11117-016-0401-9
[26] Popov, M., Randrianantoanina, B.: Narrow Operators on Function Spaces and Vector Lattices. De Gruyter Studies in Mathematics, vol. 45. De Gruyter, Berlin (2013) · Zbl 1258.47002 · doi:10.1515/9783110263343
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