## On orthogonally additive operators in C-complete vector lattices

BANACH JOURNAL OF MATHEMATICAL ANALYSIS, vol.16, no.1, 2022 (SCI-Expanded)

• Publication Type: Article / Article
• Volume: 16 Issue: 1
• Publication Date: 2022
• Doi Number: 10.1007/s43037-021-00158-2
• Journal Name:
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Business Source Elite, Business Source Premier, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
• Keywords: Orthogonally additive operator, Urysohn operator, Nemytskii operator, Lateral projection, Narrow operator, C-complete vector lattice, Lateral band, NARROW
• Hacettepe University Affiliated: Yes

#### Abstract

In this article, we investigate orthogonally additive (nonlinear) operators on C-complete vector lattices which strongly includes all Dedekind complete vector lattices. In the first part of the paper, we present basic examples of orthogonally additive operators on function spaces. Then we show that an orthogonally additive map defined on a lateral band of a C-complete vector lattice and taking values in a Dedekind complete vector lattice could be extended to the whole space and an extended orthogonally additive operator preserves continuity, narrowness, compactness and disjointness. In the second part of the article, we consider lateral projection operators onto lateral bands. One of our main results asserts that for a C-complete vector lattice E there is a lateral projection onto every lateral band of E. Applying the technique of lateral projections we prove that for a orthogonally additive narrow operator T : E -> F from a C-complete vector lattice E to an order continuous Banach lattice F all elements of the order interval [0, T] are narrow operators as well. Finally we show that T + S is a narrow operator provided that the operator T is horizontally-to-norm continuous and C-compact and the operator S is narrow.