On the Sum of Narrow and Finite-Rank Orthogonally Additive Operators

It is well known that the sum of two linear continuous narrow operators in the spaces Lp with 1 < p < ∞ is not necessarily a narrow operator. However, the sum of a narrow operator and a compact linear continuous operator is a narrow operator. In a recent paper, Pliev and Popov originated the investigation of nonlinear narrow operators and, in particular, of orthogonally additive operators. As our main result, we prove that the sum of a narrow orthogonally additive operator and a finite-rank laterally-to-norm continuous orthogonally additive operator acting from an atomless Dedekind complete vector lattice into a Banach space is a narrow operator.