We consider a positively definite self-adjoint operator $P$ on a separable Hilbert space $H$ which has a compact resolvent. Then a specific example of the Ikehara Tauberian theorem is extended to the case where the zeta function of $P$ only has simple poles. In such circumstances, we can obtain the asymptotic behavior of the counting function of eigenvalues with remainder terms. And we have their applications to some partial differential operators.
"On an Extension of the Ikehara Tauberian Theorem II." Tokyo J. Math. 18 (1) 91 - 110, June 1995. https://doi.org/10.3836/tjm/1270043611