On a Weyl inequality of operators in Banach spaces

Let s = (sn) be an injective and surjective s-number sequence in the sense of Pietsch. We show for a Riesz-operator T: X -X acting on a (complex) Banach space the following Weyl inequality between geometric means of eigenvalues and s-numbers: For any 0 1 is an absolute constant. The proof rests on an elementary mixing multiplicativity of an arbitrary s-number sequence and a striking result of G. Pisier. The inequality is a contribution to the problem of estimating eigenvalues by s-numbers first started in a strong sense by H. K6nig (1986, 2001).