The Existence of Eigenvalues for Integral Operators!1)

Throughout K(x, s) is assumed to be real valued and continuous for A x A (A=[a, i>] is a finite closed interval of the real line). For definiteness we fix the domain 2(T) of the integral operator (1) as the Hubert 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 S>(T) = LP(A), 1 z% p z% oo. It is well known that T is completely continuous. Therefore, the eigenvalues A = {X0,Xy,X2,---} 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 {X,} 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.,