On sums of narrow operators on Kِthe function spaces

We find necessary and sufficient conditions on a Köthe Banach space E on [ 0 , 1 ] and a Banach space X under which a sum of two narrow operators from E to X is narrow. Using this condition, we prove that, given a Köthe Banach space E on [ 0 , 1 ] , there exist a Banach space X and narrow operators T 1 , T 2 : E → X with non-narrow sum T = T 1 + T 2 . In particular, this answers in the negative, a question of V.M. Kadets and the second named author, of whether for every Banach space X a sum of two narrow operators from L 1 to X must be narrow. Another result asserts that for every 1 < p ≤ ∞ there are regular narrow operators T 1 , T 2 : L p → L ∞ with non-narrow sum T = T 1 + T 2 . For p = ∞ this answers a question of O.V. Maslyuchenko and the authors.