Some problems on narrow operators on function spaces

Abstract

It is known that if a rearrangement invariant (r.i.) space E on [0 1] has an unconditional basis then every linear bounded operator on E is a sum of two narrow operators. On the other hand, for the classical space E = L 1 [0 1] having no unconditional basis the sum of two narrow operators is a narrow operator. We show that a Köthe space on [0 1] having "lots" of nonnarrow operators that are sum of two narrow operators need not have an unconditional basis. However, we do not know if such an r.i. space exists. Another result establishes sufficient conditions on an r.i. space E under which the orthogonal projection onto the closed linear span of the Rademacher system is a hereditarily narrow operator. This, in particular, answers a question of the first named author and Randrianantoanina (Problem 11.9 in [Popov M., Randrianantoanina B., Narrow Operators on Function Spaces and Vector Lattices, de Gruyter Stud. Math., 45, Walter de Gruyter, Berlin, 2013]). MSC:46B42, 47B38

Narrow orthogonally additive operators

“…For example, a sum of two narrow operators on L 1 is narrow, but on an r.i. space E on [0, 1] with an unconditional basis every operator is a sum of two narrow operators. Similar questions are very interesting and some of them are involved, see problems [18,Chapter 12] and recent papers [11], [17]. In 2009 O. Maslyuchenko, Mykhaylyuk and the second named author discovered that the sum phenomenon for narrow operators has a pure lattice nature, extending the notion to vector lattices [10].…”

Optimal range of Haar martingale transforms and its applications

“…p.161]). The main result in this subsection is linked with the following open problem stated in [27]:…”