Essential and discrete spectra of partially integral operators

Let $\Omega_1,\Omega_2\subset\mathbb R^\nu$ be compact sets. In the Hilbert space $L_2(\Omega_1\times\Omega_2)$, we study the spectral properties of selfadjoint partially integral operators $T_1$, $T_2$, and $T_1+T_2$, with
\begin{align*} (T_1 f)(x,y)&=\int_{\Omega_1}k_1(x,s,y)f(s,y)d\mu(s), \\ (T_2 f)(x,y)&=\int_{\Omega_2}k_2(x,t,y)f(x,t)d\mu(t), \end{align*}
whose kernels depend on three variables. We prove a theorem describing properties of the essential and discrete spectra of the partially integral operator $T_1+T_2$.