Some stability theorems on narrow operators acting in $L_1$ and $C(K)$

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.