Spectra of partial integral operators with a kernel of three variables

Let Ω= [a, b] × [c, d] and T1, T2 be partial integral operators in \( C \)(Ω): (T1f)(x, y) = \( \mathop \smallint \limits_a^b \)k1(x, s, y)f(s, y)ds, (T2f)(x, y) = \( \mathop \smallint \limits_c^d \)k2(x, ts, y)f(t, y)dt where k1 and k2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT1, τ ∈ ℂ and E−τT2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T1, T2, and T = T1 + T2 are proved.