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

In this paper, we consider linear bounded self-adjoint integral operators T1 and T2 in the Hilbert space L2([a,b]×[c,d]), the so-called partially integral operators. The partially integral operator T1 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 T2 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 T1T2 is compact and T1T2=T2T1. 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 T1, T2 and T1+T2 with nondegenerate kernels are investigated. A formula is obtained for describing the essential spectra of the partially integral operators T1 and T2. It is shown that the operators T1 and T2 have no discrete spectrum. A theorem on the structure of the essential spectrum of the partially integral operator T1+T2 is proved. The problem of the existence of a countable number of eigenvalues in the discrete spectrum of the partially integral operator T1+T2 is studied.