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This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices-orthosymmetric maps-, C *-algebras-zero product preserving operators-, and classical and harmonic analysis-integral bilinear operators-. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

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