Abstract: In this paper, we consider linear bounded self-adjoint integral operators \(T_1\) and \(T_2\) in the Hilbert space \(L_2([a,b]\times[c,d])\), the so-called partially integral operators. The partially integral operator \(T_1\) acts on the functions \(f(x,y)\) with respect to the first argument and performs a certain integration with respect to the argument \(x\), and the partially integral operator \(T_2\) acts on the functions \(f(x,y)\) with respect to the second argument and performs some integration over the argument \(y\). Both operators are bounded, however both are not compact operators. However, the operator \(T_1T_2\) is compact and \(T_1T_2=T_2T_1\). Partially integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrodinger operators. In this paper, the spectral properties of linear bounded self-adjoint partially integral operators \(T_1\), \(T_2\) and \(T_1+T_2\) with nondegenerate kernels are investigated. A formula is obtained for describing the essential spectra of the partially integral operators \(T_1\) and \(T_2\). It is shown that the operators \(T_1\) and \(T_2\) have no discrete spectrum. A theorem on the structure of the essential spectrum of the partially integral operator \(T_1+T_2\) is proved. The problem of the existence of a countable number of eigenvalues in the discrete spectrum of the partially integral operator \(T_1+T_2\) is studied.
For citation: Kulturayev, D. J. and Eshkabilov, Yu. Kh. Spectral Properties of Self-Adjoint Partially Integral Operators with Non-Degenerate Kernels, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp. 91-104 (in Russian). DOI 10.46698/y9559-5148-4454-e
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