Abstract
We consider bounded selfadjoint linear integral operators \( T_{1} \) and \( T_{2} \) in the Hilbert space \( L_{2}([a,b]\times[c,d]) \) which are usually called partial integral operators. We assume that \( T_{1} \) acts on a function \( f(x,y) \) in the first argument and performs integration in \( x \), while \( T_{2} \) acts on \( f(x,y) \) in the second argument and performs integration in \( y \). We assume further that \( T_{1} \) and \( T_{2} \) are bounded but not compact, whereas \( T_{1}T_{2} \) is compact and \( T_{1}T_{2}=T_{2}T_{1} \). Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of \( T_{1} \), \( T_{2} \), and \( T_{1}+T_{2} \) with nondegenerate kernels and established some formula for the essential spectra of \( T_{1} \) and \( T_{2} \). Furthermore, we demonstrate that the discrete spectra of \( T_{1} \) and \( T_{2} \) are empty, and prove a theorem on the structure of the essential spectrum of \( T_{1}+T_{2} \). Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of \( T_{1}+T_{2} \).
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Translated from Vladikavkazskii Matematicheskii Zhurnal, 2022, Vol. 24, No. 4, pp. 91–104. https://doi.org/10.46698/y9559-5148-4454-e
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Kulturaev, D.Z., Eshkabilov, Y.K. On the Spectral Properties of Selfadjoint Partial Integral Operators with a Nondegenerate Kernel. Sib Math J 65, 475–486 (2024). https://doi.org/10.1134/S0037446624020204
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DOI: https://doi.org/10.1134/S0037446624020204