Geodesic
Some stability theorems on narrow operators acting in $L_1$ and $C(K)$
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003) no. 1, pp. 49-60 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new proof of two stability theorems concerning narrow operators acting from $L_1$ to $L_1$ or from $C(K)$ to an arbitrary Banach space is given. Namely a sum of two such operators and moreover a sum of a point-wise unconditionally convergent series of such operators is a narrow operator again. The relations between several possible definitions of narrow operators on $L_1$ are also discussed. 
@article{JMAG_2003_10_1_a4,
     author = {V. M. Kadets and M. M. Popov},
     title = {Some stability theorems on narrow operators acting in $L_1$ and~$C(K)$},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {49--60},
     year = {2003},
     volume = {10},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2003_10_1_a4/}
}
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V. M. Kadets; M. M. Popov. Some stability theorems on narrow operators acting in $L_1$ and $C(K)$. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003) no. 1, pp. 49-60. http://geodesic.mathdoc.fr/item/JMAG_2003_10_1_a4/