Narrow operators on tensor products of Köthe spaces

Abstract

We investigate narrow operators on tensor products of Köthe-Banach spaces. Our first main result asserts that for a bounded linear operator S:F→F on Köthe-Banach space F with an order continuous norm and a linear bounded operator T:L1(μ)→L1(μ), the narrowness at least one of the T or S implies the narrowness for an operator T⊗S:L1(μ)⊗ˆπF→L1(μ)⊗ˆπF defined on the positive projective tensor product of L1(μ) 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 Köthe-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 Köthe-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.