On partial integral representation of linear positive operators

In this paper, we obtain a criterion for partial integral representability of positive L∞-homogeneous operators acting in ideal spaces of measurable real functions defined on the product of measurable spaces with σ-finite measures. The result obtained is a counterpart of Bukhvalov's criterion for integral representability of linear operators acting in ideal spaces of measurable real functions defined on measurable spaces with σ-finite measures. Note that under certain conditions, the above-mentioned Bukhvalov criterion can be derived from the result obtained in this paper. Consequently, the result obtained is a generalization of Bukhvalov's criterion. The main tools of this study are the above-mentioned Bukhvalov criterion and the methods of vector lattice theory.