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 andperforms 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 operatorsarise in various areas of mechanics, the theory of integro-differential equations,and the theory of Schrödinger operators. We study the spectral propertiesof $ T_{1} $ , $ T_{2} $ , and $ T_{1}+T_{2} $ with nondegenerate kernelsand 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 eigenvaluesin the discrete spectrum of $ T_{1}+T_{2} $ .
