Spectra of partial integral operators with a kernel of three variables

Let Omega = [a, b] x [c, d] and T(1), T(2) be partial integral operators in C(Omega) : (T(1)f)(x, y) = integral(b)(a) k(1)( x, s, y)f(s, y) d s; (T(2)f)(x, y)= integral(d)(c) k(2)(x, ts, y)f(t, y) d t where k(1) and k(2) are continuous functions on [a, b] x Omega and Omega x [c, d], respectively. In this paper, concepts of determinants and minors of operators E-tau T(1), tau is an element of C and E-tau T(2), tau is an element of C are introduced as continuous functions on [a, b] and [c, d]; respectively. Here E is the identical operator in C(Omega) : In addition, Theorems on the spectra of bounded operators T(1), T(2), and T = T(1) + T(2) are proved.