Abstract: In this paper, we obtain a criterion for partial integral representability of positive \(L^\infty\)-homogeneous operators acting in ideal spaces of measurable real functions defined on the product of measurable spaces with \(\sigma\)-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 \(\sigma\)-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.
Keywords: ideal space, partial integral operator, positive operator, integral operator
For citation: Orinbaev, P. R. and Tasoev, B. B. On Partial Integral Representation of Linear Positive Operators, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 101-111 (in Russian). DOI 10.46698/s1056-5701-7829-j
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