DOMINATED ORTHOGONALLY ADDITIVE OPERATORS IN LATTICE-NORMED SPACES

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 vertical bar Tx vertical bar <= S vertical bar x vertical bar 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.