It is well known that the sum of two linear continuous narrow operators in the spaces Lp with 1 < p < ∞ is not necessarily a narrow operator. However, the sum of a narrow operator and a compact linear continuous operator is a narrow operator. In a recent paper, Pliev and Popov originated the investigation of nonlinear narrow operators and, in particular, of orthogonally additive operators. As our main result, we prove that the sum of a narrow orthogonally additive operator and a finite-rank laterally-to-norm continuous orthogonally additive operator acting from an atomless Dedekind complete vector lattice into a Banach space is a narrow operator.
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References
C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht (2006).
V. Mykhaylyuk, M. Pliev, and M. Popov, Laterally Ordered Sets and Orthogonally Additive Operators, Preprint (2015).
A. I. Gumenchuk, “Lateral continuity and orthogonally additive operators,” Carpath. Math. Publ. (2015).
J. M. Mazón and S. Segura de León, “Order bounded orthogonally additive operators,” Rev. Roum. Math. Pures Appl., 35, No. 4, 329–353 (1990).
J. M. Mazón and S. Segura de León, “Uryson operators,” Rev. Roum. Math. Pures Appl., 35, No. 5, 431–449 (1990).
A. G. Kusraev and M. A. Pliev, “Orthogonally additive operators on lattice-normed spaces,” Vladikavkaz Math. J., No. 3, 33–43 (1999).
A. G. Kusraev and M. A. Pliev, “Weak integral representation of the dominated orthogonally additive operators,” Vladikavkaz Math. J., No. 4, 22–39 (1990).
M. Pliev, “Uryson operators on the spaces with mixed norm,” Vladikavkaz Math. J., No. 3, 47–57 (2007).
A. I. Gumenchuk, M. A Pliev, and M. M. Popov, “Extensions of orthogonally additive operators,” Math. Stud., 41, No. 2, 214–219 (2014).
A. M. Plichko and M. M. Popov, “Symmetric function spaces on atomless probability spaces,” Diss. Math., 306, 1–85 (1990).
M. M. Popov, “Narrow operators (a survey),” Banach Center Publ., 92, 299–326 (2011).
M. Popov and B. Randrianantoanina, Narrow Operators on Function Spaces and Vector Lattices, De Gruyter, Berlin (2013).
V. Mykhaylyuk, “On the sum of a compact and a narrow operators,” J. Funct. Anal., 266, 5912–5920 (2014).
O. V. Maslyuchenko, V. V. Mykhaylyuk, and M. M. Popov, “A lattice approach to narrow operators,” Positivity, 13, 459–495 (2009).
M. A. Pliev and M. M. Popov, “Narrow orthogonally additive operators,” Positivity, 18, 641–667 (2014).
V. M. Kadets and M. I. Kadets, Rearrangements of Series in Banach Spaces, American Mathematical Society, Providence, RI (1991).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 12, pp. 1620–1625, December, 2015.
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Humenchuk, H.I. On the Sum of Narrow and Finite-Rank Orthogonally Additive Operators. Ukr Math J 67, 1831–1837 (2016). https://doi.org/10.1007/s11253-016-1193-6
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DOI: https://doi.org/10.1007/s11253-016-1193-6