Abstract: We consider narrow linear operators defined on a Banach-Kantorovich space and taking value in a Banach space. We prove that the sum \(S+T\) of two operators is narrow whenever \(S\) is a narrow operator and \(T\) is a \((bo)\)-continuous \(C\)-compact operator. For the proof of the main result we use the method of decomposition of an element of a lattice-normed space into a sum of disjoint fragments and an approximation of a \(C\)-com\-pact operator by finite-rank operators.
For citation: Abasov N. M., Pliev M. A. On the Sum of Narrow and \(C\)-Compact Operators. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp.3-9. DOI 10.23671/VNC.2018.1.11391
1. Popov M. M., Plichko A. M. Symmetric function spaces on atomless
probability spaces: Diss. Math. (Rozprawy Mat.), 1990, vol. 306. pp.
1-85.
2. Popov M., Randrianantoanina B. Narrow Operators on Function
Spaces and Vector Lattices, De Gruyter, 2013 (De Gruyter Stud. in
Math., vol. 45).
3. Maslyuchenko O., Mykhaylyuk V., Popov M. A lattice approach to
narrow operators, Positivity, 2009, vol. 13, pp. 459-495. DOI:
10.1007/s11117-008-2193-z.
4. Pliev M. Narrow operators on lattice-normed spaces, Open Math.,
2011, vol. 9, no. 6, pp. 1276-1287. DOI: 10.2478/s11533-011-0090-3.
5. Abasov N., Megahed A. M., Pliev M. Dominated operators from
lattice-normed spaces to sequence Banach lattices, Annals of Funct.
Anal., 2016, vol. 7, no. 4, pp. 646-655. DOI:
10.1215/20088752-3660990.
6. Pliev M., Popov M. Narrow orthogonally additive operators,
Positivity, 2014, vol. 18, no. 4, pp. 641-667. DOI:
10.1007/s11117-013-0268-y.
7. Mykhaylyuk V., Pliev M., Popov M., and Sobchuk O. Dividing
measures and narrow operators, Stud. Math., 2015, vol. 231, pp.
97-116. DOI: 10.4064/sm7878-2-2016.
8. Pliev M. Domination problem for narrow orthogonally additive
operators, Positivity, 2017, vol. 21, no. 1, pp. 23-33. DOI:
10.1007/s11117-016-0401-9.
9. Pliev M. A., Fang X. Narrow orthogonally additive operators in
lattice-normed spaces, Siberian Math. J., 2017, vol. 58, no. 1, pp.
134-141. DOI: 10.1134/S0037446617010177.
10. Mykhaylyuk V., Popov M. On sums of narrow operators on Kothe
function space, J. Math. Anal. Appl., 2013, vol. 404, pp. 554-561.
DOI: 10.1016/j.jmaa.2013.03.008.
11. Humenchuk H. I. On the sum of narrow and finite-rank
orthogonally additive operator, Ukrainian Math. J., 2016, vol. 67,
no. 12, pp. 1831-1837. DOI: 10.1007/s11253-016-1193-6.
12. Kusraev A. G. Dominated Operators, Dordrecht-Boston-London:
Kluwer Acad. Publ., 2000, 446 p.
