Essential and discrete spectra of partially integral operators

Let Ω1, Ω2 ⊂ ℝν be compact sets. In the Hilbert space L 2(Ω1 × Ω2), we study the spectral properties of selfadjoint partially integral operators T 1, T 2, and T 1 + T 2, with $$ \begin{gathered} (T_1 f)(x,y) = \int_{\Omega _1 } {k_1 (x,s,y)f(s,y)d\mu (s),} \hfill \\ (T_2 f)(x,y) = \int_{\Omega _2 } {k_2 (x,t,y)f(x,t)d\mu (t),} \hfill \\ \end{gathered} $$ 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.

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