Dominated orthogonally additive operators in lattice-normed spaces

Document Type: Special issue: Trends in Operators on Banach Spaces

Authors

1 ‎Moscow Aviation Institute, Russia

2 ‎Southern Mathematical Institute of the Russian Academy of Sciences‎, ‎Russia

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 $\ls Tx \rs\leq S\ls‎
‎x\rs$ 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.

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