On the sum of narrow orthogonally additive operators

N. M. Abasov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2020), pp. 3-9

Résumé

In this article we consider orthogonally additive operators defined on a vector lattice $E$ and taking value in a Banach space $X$. We say that an orthogonally additive operator $T:E\to X$ is a narrow if for every $e\in E$ and $\varepsilon>0$ there exists a decomposition $e=e_1\sqcup e_2$ of $e$ into a sum of two disjoint fragments $e_1$ and $e_2$ such that $\|Te_1-Te_2\|\varepsilon$. It is proved that the sum of two orthogonally additive operators $S+T$ defined on Dedekind complete, atomless vector lattice and taking value in Banach space, where $S$ is a narrow operator and $T$ is a $C$-compact laterally-to-norm continuous operator, is a narrow operator as well.

Détail

Keywords: vector lattice, orthogonally additive operator, narrow operator, laterally-to-norm continuous operator, $C$-compact operator.