On sums of narrow operators on Köthe function spaces

We find necessary and sufficient conditions on a Köthe Banach space EE on [0,1][0,1] and a Banach space XX under which a sum of two narrow operators from EE to XX is narrow. Using this condition, we prove that, given a Köthe Banach space EE on [0,1][0,1], there exist a Banach space XX and narrow operators T1,T2:E→XT1,T2:E→X with non-narrow sum T=T1+T2T=T1+T2. In particular, this answers in the negative, a question of V.M. Kadets and the second named author, of whether for every Banach space XX a sum of two narrow operators from L1L1 to XX must be narrow. Another result asserts that for every 1