ABSTRACT
Mz(t,s) = Is m(s,u)z(t,u)dll(u). In general, even in case of continuous kernels the equation (15.27) is not Fredholm, because it contains the partial integral operators L and M. However, if the operators L and M are continuous in some Banach space X, and N is compact in X, in some important cases one may prove that the integral equation (15.27) is Fredholm. This is possible if the operators L and M may be represented as tensor products L = L®I and M = I®M, where L and M are defined by (14.3) and (14.4). If we use now the results of § 14 and the fact that the Fredholm property and the index of an operator is stable with respect to compact perturbations (see e.g. KATO [1966] or KREJN [1971]), we arrive at the following results. Theorem 15.6. Suppose that the integral operator L acts in an ideal space U = U(T), the integral operator M acts in an ideal space V = V(S), and the integral operator N is compact in the space X = [U ..... VI or X = [U - VI. Assume that one of the following conditions is satisfied: (a) U = Lp(T) and V = Lp(S) (1 S P < (0); (b) V = LI(S), U is almost perfect, and X = [U ..... VI; (c) U =LI(T), V is almost perfect, and X = [U +- VI. Then the integral equation (15.27) is Fredholm of indez zero if and only if 1 ¢ ue,(X), and Fredholm if and only if 1 ¢ un/(X); here ue,(X) and un/(X) are the sets defined in Theorem 1-1.1. In case 1 ¢ un/(X), the index of equation (15.27) may be calculated by formula (14.10).
