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  \( T_{1}+T_{2} \) .