On the Spectral Properties of Selfadjoint Partial Integral Operators with a Nondegenerate Kernel

Abstract

We consider bounded selfadjoint linear integral operators $T_{1}$ and $T_{2}$ in the Hilbert space $L_{2}([a,b]\times[c,d])$ which are usually called partial integral operators. We assume that $T_{1}$ acts on a function $f(x,y)$ in the first argument and performs integration in $x$, while $T_{2}$ acts on $f(x,y)$ in the second argument and performs integration in $y$. We assume further that $T_{1}$ and $T_{2}$ are bounded but not compact, whereas $T_{1}T_{2}$ is compact and $T_{1}T_{2}=T_{2}T_{1}$. Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of $T_{1}$, $T_{2}$, and $T_{1}+T_{2}$ with nondegenerate kernels and established some formula for the essential spectra of $T_{1}$ and $T_{2}$. Furthermore, we demonstrate that the discrete spectra of $T_{1}$ and $T_{2}$ are empty, and prove a theorem on the structure of the essential spectrum of $T_{1}+T_{2}$. Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of $T_{1}+T_{2}$.