Abstract
In this paper, we study partial integral operators on Banach–Kantorovich spaces over a ring of measurable functions. We obtain a decomposition of the cyclic modular spectrum of a bounded modular linear operator on a Banach–Kantorovich space in the form of a measurable bundle of spectra of bounded operators on Banach spaces. The classical Banach spaces with mixed norm are endowed with the structure of Banach–Kantorovich modules. We use such representations to show that every partial integral operator on a space with a mixed norm can be represented as a measurable bundle of integral operators. In particular, we show the cyclic compactness of such operators and, as an application, prove the Fredholm \(\nabla\)-alternative. We also give an example of a partial integral operator with a nonempty cyclically modular discrete spectrum, while its modular discrete spectrum is an empty set.
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Acknowledgments
The authors thank the referee for valuable remarks and constructive suggestions that have helped to improve the quality of the presented investigation and pretty much organize the paper following the requirements of the journal.
Funding
This work of the second author was supported by the Ministry of Science and Higher Education of the Russian Federation (grant no. 075-02-2022-896).
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Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 18–37 https://doi.org/10.4213/mzm13703.
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Arziev, A.D., Kudaybergenov, K.K., Orinbaev, P.R. et al. Partial Integral Operators on Banach–Kantorovich Spaces. Math Notes 114, 15–29 (2023). https://doi.org/10.1134/S0001434623070027
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DOI: https://doi.org/10.1134/S0001434623070027