The existence of eigenvalues for integral operators

Throughout K(x, s) is assumed to be real valued and continuous for A x A (A = [a, b] is a finite closed interval of the real line). For definiteness we fix the domain 9(T) of the integral operator (1) as the Hilbert space L2(A); it should be emphasized at this point that K(x, s) is not necessarily a symmetric kernel. It will be clear from the subsequent analysis that the nature of the spectral set of T is unaltered for any of the alternative specifications g(T)= Lp(A), 1 < p < 00. It is well known that T is completely continuous. Therefore, the eigenvalues A = {)'o2, ...} form a discrete set which may be infinite, finite or empty. Each eigenvalue is of finite algebraic and geometric multiplicity and 0 is the only limit point of {,A} if A is not finite. Finally, the spectrum of the transformation T, apart from point spectrum A, can contain only the origin. Let r(T) denote the spectral radius of T, i.e.,