Abstract: The problem of describing linear operators representable as integral operators was posed by J. von Neumann in the mid-1930s and for a long time remained one of the central problems in operator theory and functional analysis. A significant contribution to its solution was made in 1974 by Buchvalov, who established a criterion for the integral representability of linear operators in ideal function spaces. In subsequent studies, this topic has been further developed: in a recent work by Orynbayev and Tasoev, a criterion for partial integral representability of positive \(L_\infty\)-homogeneous operators on sigma-finite spaces was obtained. In the present paper, a new notion of modular equimeasurability is introduced, based on the concept of cyclic compactness. Using this approach, it is proved that every partially integral operator acting in Banach ideal function spaces maps order intervals into modularly equimeasurable sets, which significantly extends and generalizes previously known results in this area.
Keywords: partial integral operator, integral operator, Banach ideal function spaces.
Fund name: The study was supported by a grant from the Russian Science Foundation, project No. 24-71-10094, https://rscf.ru/project/24-71-10094/.
For citation: Kudaybergenov, K. K. and Orinbaev, P. R. Partial Integral Operators in Banach Ideal Function Spaces, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 73-81(in Russian). DOI 10.46698/i0132-3339-6227-v
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