Narrow operators on tensor products of Kothe spaces

We investigate narrow operators on tensor products of Kothe-Banach spaces. Our first main result asserts that for a bounded linear operator S : F -> F on Kothe-Banach space F with an order continuous norm and a linear bounded operator T : L-1(mu) -> L-1(mu), the narrowness at least one of the T or S implies the narrowness for an operator T circle times S : L1(mu)(circle times) over cap F-pi -> L-1(mu)(circle times) over cap F-pi defined on the positive projective tensor product of L1(mu) and F. The converse statement holds under an additional assumption of regularity for the operator S : F -> F. As a consequence, we resolve an open problem concerning the existence of a Kothe-Banach space on [0, 1] with an absolutely continuous norm without an unconditional basis in which the identity operator is a sum of two narrow operators, suggested in [45]. We also supply complementing results for the Kothe-Banach spaces E[X] with mixed norm where E and X are symmetric spaces and fully characterize the spaces E[X] having an unconditional basis. (c) 2022 Elsevier Inc. All rights reserved.