The existence of eigenvalues for integral operators
Samuel Karlin · Transactions of the American Mathematical Society · 1964
In the following paper we establish conditions for the existence of an infinite, simple point spectrum (and properties of the corresponding eigenfunctions {cp,,}) for the integral operator (1) (T ] 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., (2) r(T)= max|A¡|.For X>r(T) the Neumann expansion applies (convergence is understood in the sense of the operator norm) :The iterated operator Tm is associated with the iterated kernel Kin)(x, s) defined by (4) K(n)(x,S)= f ».Í K(x,Sy)K(Sy,S2)-K(Sn-y,S)dSy-dSn-y.Já Jà Specifically,