Narrow operators on [formula omitted]-complete lattice-normed spaces

Abstract

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.