Dominated orthogonally additive operators in lattice-normed spaces

Abstract

‎In this paper, we introduce a new class of operators in lattice-normed spaces‎. ‎We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space‎ ‎$(W,F)$ is dominated‎, ‎if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $\vert Tx \vert \leq S \vert x \vert$ for any element $x$ of $(V,E)$‎. ‎We show that under some mild‎ ‎conditions‎, ‎a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator‎. ‎In the last part of the‎ ‎paper we consider laterally-to-order continuous operators‎. ‎We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant‎.