I know that if an operator (on a Banach space with approximation property) is nuclear of order zero, then its eigenvalues are $p$-summable for any $p>0$. (I read it from Grothendieck’s book “Produits tensoriels topologiques et espaces nucléaires” II. P 16. Theorem 4, if I didn’t misunderstand the statement.) I wonder if the converse is true?

Or in general, is there any results on deducing the convergence rate of ”singular values” from the convergence rate of eigenvalues?