Spectral properties of self-adjoint partially integral operators with non-degenerate kernels

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.