Spectral properties of self-adjoint partially integral operators with non-degenerate kernels
D. J. Kulturayev,
Yu. Kh. Eshkabilov Karshi State University, 17 Kuchabag St., Karshi 180100, Uzbekistan
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.
Key words:
partially integral operator, spectra, essential spectrum, discrete spectrum, non-degenerate kernel.
UDC:
517.984.46
MSC: 47A10,
47A11,
47B38,
47G10 Received: 19.10.2021
DOI:
10.46698/y9559-5148-4454-e
Fulltext:
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