Orthogonally biadditive operators (Q2064416)

Summary: In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator \(T:E\times F\longrightarrow W\) defined on a Cartesian product of vector lattices \(E\) and \(F\) and taking values in a vector lattice \(W\) is orthogonally biadditive if all partial operators \(T_y:E\longrightarrow W\) and \(T_x:F\longrightarrow W\) are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators \(\mathscr{OBA}_r(E, F; W)\) is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in \(\mathscr{OBA}_r(E, F; W)\). In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.