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

1. Vekua, I. N. Novye metody resheniya ellipticheskikh uravneniy
[New Methods for Solving Elliptic Equations], Moscow, Nauka, 1948, 296 p.
(in Russian).

2. Aleksandrov, V. M. and Kovalenko, E. V. On a Class of Integral Equations in Mixed
Problems of Continum Mechanics, Soviet Physics. Doklady, 1980, vol. 25, no. 2, pp. 354-356.

3. Aleksandrov, V. M. and Kovalenko, E. V. Contact Interaction of Bodies with Coatings
in the Presense of Abrasion, Soviet Physics. Doklady, 1984, vol. 29, no. 4, pp. 340-342.

4. Manzhirov, A. V. On a Method for Solving Two-Dimensional
Integral Equation for Exially Symmetric Contact Problem for Bodies
with Complex Layer Rheology, Journal of Applied Mathematics and Mechanics,
1985, vol. 49, no. 6, pp. 777-782. DOI: 10.1016/0021-8928(85)90016-4.

5. Gursa, E. Kurs matematicheskogo analiza. T. 3, ch. 2
[A Course of Mathematical Analysis. Vol. 3, part 2], Moscow, Leningrad,
1934, 318 p. (in Russian).

6. Muntz, G. Integral'nye uravneniya. T. 1 [Integral Equations. Vol. 1],
Leningrad, Moscow, 1934, 330 p. (in Russian).

7. Eshkabilov, Yu. Kh. A Discrete "Three-Particle" Operator Schrodinger
in the Hubbard Model, Theoretical and Mathematical Physics, 2006, vol. 149, no. 2,
pp. 1497-1511. DOI: 10.1007/s11232-006-0133-2.

8. Albeverio, S., Lakaev, S. N. and Muminov, Z. I. On the Number of Eigenvalues
of a Model Operator Associated to a System of Three-Particles on Lattices,
Russian Journal of Mathematical Physics, 2007, vol. 14, no. 4, pp. 377-387.
DOI: 10.1134/S1061920807040024.

9. Rasulov, T. Kh. Asymptotics of the Discrete Spectrum of One
Model Operator Associated with a System of Three Particles on
Lattice, Theoretical and Mathematical Physics, 2010, vol. 163, no. 1, pp. 429-437.
DOI: 10.1007/s11232-010-0033-3.

10. Appell, J. M, Kalitvin, A. S. and Nashed, M. Z. On Some Partial Integral
Equations Arising in the Mechanics of Solids,
Journal of Applied Mathematics and Mechanics,
1999, vol. 79, no. 10, pp. 703-713.
DOI: 10.1002/(SICI)1521-4001(199910)79:10<703::AID-ZAMM703>3.0.CO;2-W.

11. Kalitvin, A. S. Lineynye operatory s chastnymi integralami
[Linear Operators with Partial Integrals], Voronezh, Central Black Earth
Book Publishing House, 2000, 252 p. (in Russian).

12. Appell, J. M., Kalitvin, A. S. and Zabrejko, P. P. Partial Integral Operators
and Integro-Differential Equations, New York, 2000, 578 p. DOI: 10.1201/9781482270402.

13. Kalitvin, A. S. On the Spectrum of Linear Operators with Partial Integrals and Positive Kernels,
Operatory i ikh prilozheniya: Mezhvuzovskiy sbornik nauchnukh trudov
[Operators and their Applications: Interuniversity Compilation of Scientific Works], Leningrad, 1988, pp. 43-50 (in Russian).

14. Kalitvin, A. S. and Zabrejko, P. P. On the Theory of Partial Integral Operators,
Journal of Integral Equations and Applications, 1991, vol. 3, no. 3, pp. 351-382.
DOI: 10.1216/jiea/1181075630.

15. Kalitvin, A. S. and Kalitvin, V. A. Linear Operators and Equations with Partial Integrals,
Contemporary Mathematics. Fundamental Directions, 2019, vol. 65, no. 3, pp. 390-433 (in Russian).
DOI: 10.22363/2413-3639-2019-65-3-390-433.

16. Eshkabilov, Yu. Kh. On the Spectrum of the Tensor Sum of Compact Operators,
Uzbek Mathematical Journal, 2005, no. 3, pp. 104-112 (in Russian).

17. Eshkabilov, Yu. Kh. Partial Integral Operator with Bounded
Kernels, Siberian Advances in Mathematics, 2008, vol. 19, no. 3, pp. 151-161.
DOI: 10.3103/S1055134409030018.

18. Eshkabilov, Yu. Kh. Essential and Discrete Spectra of Partially Integral Operators,
Siberian Advances in Mathematics, 2008, vol. 19, no. 4, pp. 233-244.
DOI: 10.3103/S1055134409040026.

19. Eshkabilov, Yu. Kh. On the Discrete Spectrum of Partially Integral Operators,
Siberian Advances in Mathematics, 2012, vol. 23, no. 4, pp. 227-233.
DOI: 10.3103/S1055134413040019.

20. Arzikulov, G. P. and Eshkabilov, Yu. Kh. On the Essential and the Discrete Spectra
of a Fredholm Type Partial Integral Operator, Siberian Advances in Mathematics,
2014, vol. 25, no. 4, pp. 231-242. DOI: 10.3103/S105513441504001X.

21. Arzikulov, G. P. and Eshkabilov, Yu. Kh. On the Spectra of Partial Integral Operators,
Uzbek Mathematical Journal, 2015, no. 2, pp. 148-159.

22. Reed, M. and Simon, B. Metody sovremennoy matematicheskoy fiziki. T. 1. Funktsional'nyy analiz
[Methods of Modern Mathematical Physics. Vol. 1. Functional Analysis],
Ìoscow, Mir, 1977, 412 p. (in Russian).

23. Pankrashkin, K. Introduction to the Spectral Theory, Orsay, 2014.

23. Kantorovich, L. V. and Akilov, G. P. Funktsional'nyy analiz
[Functional Analysis], Moscow, Nauka, 1984, 750 p. (in Russian).

24. Eshkabilov, Yu. Kh. On Infinity of the Discrete Spectrum
Operators in the Friedrichs Model, Siberian Advances in Mathematics,
2012, vol. 22, no. 1, pp. 1-12. DOI: 10.3103/S1055134412010014.