Geodesic
On the sum of narrow and $C$-compact operators
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 3-9

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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 $C$-compact operator by finite-rank operators. 
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     author = {N. M. Abasov and M. A. Pliev},
     title = {On the sum of narrow and $C$-compact operators},
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N. M. Abasov; M. A. Pliev. On the sum of narrow and $C$-compact operators. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 3-9. http://geodesic.mathdoc.fr/item/VMJ_2018_20_1_a0/