Narrow Orthogonally Additive Operators on Lattice-Normed Spaces

Abstract

The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between lattice-normed spaces. We prove that every $C$-compact laterally-to-norm continuous orthogonally additive operator from a Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. We also show that every dominated Uryson operator from Banach-Kantorovich space over an atomless Dedekind complete vector lattice $E$ to a sequence Banach lattice ${\ell }_p(\Gamma )$ or $c_0(\Gamma )$ is narrow. Finally, we prove that if an orthogonally additive dominated operator $T$ from lattice-normed space $(V,E)$ to Banach-Kantorovich space $(W,F)$ is order narrow then the order narrow is its exact dominant $\ls T\rs$.

16 pages. arXiv admin note: substantial text overlap with arXiv:1508.03275, arXiv:1309.5490

Cited by in corpus (1)

• The Dodds-Fremlin type theorem for abstract Uryson operators