On the eigenvalues and eigenfunctions of some integral operators

Some properties of the eigenvalues of the integral operator K_gt defined as K_τf(x) = ∫₀^τK(x ? y) f (y) dy were studied by Vittal Rao (J. Math. Anal. Appl.53 (1976), 554–566), with some assumptions on the kernel K(x). In this paper the eigenfunctions of the operator K_τ are shown to be continuous functions of τ under certain circumstances. Also, the results of Vittal Rao and the continuity of eigenfunctions are shown to hold for a larger class of kernels.