Abstract.
It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N r (E, F) of all narrow regular operators is a band in the vector lattice L r (E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N r (E, F) in L r (E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T D + T N where T D is a sum of an order absolutely summable family of disjointness preserving operators and T N is narrow.
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Maslyuchenko, O.V., Mykhaylyuk, V.V. & Popov, M.M. A lattice approach to narrow operators. Positivity 13, 459–495 (2009). https://doi.org/10.1007/s11117-008-2193-z
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DOI: https://doi.org/10.1007/s11117-008-2193-z