Document Type: Special issue: Trends in Operators on Banach Spaces
Moscow Aviation Institute, Russia
Southern Mathematical Institute of the Russian Academy of Sciences, Russia
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 $\ls Tx \rs\leq S\ls
x\rs$ 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.