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Linear Operators and Equations with Partial Integrals

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We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra–Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 65, No. 3, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2019.

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Kalitvin, A.S., Kalitvin, V.A. Linear Operators and Equations with Partial Integrals. J Math Sci 265, 196–235 (2022). https://doi.org/10.1007/s10958-022-06052-y

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