On a Weyl inequality of operators in Banach spaces

Let $s=(s_n)$ be an injective and surjective $s$-number sequence in the sense of Pietsch. We show for a Riesz-operator $T:X\to X$ acting on a (complex) Banach space the following Weyl inequality between geometric means of eigenvalues and $s$-numbers: For any $0<\delta \le 1$ and all $n\in\mathbb{N}$,

$$\left(\prod\limits_{i=1}^n \vert\lambda_i(T)\vert\right)^{\frac 1... ...{1+\delta}\right]} s_i(T)\right)^{\frac 1{\left[\frac n {1+\delta}\right]}}~,$$

where $c_0\ge 1$ 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. König (1986, 2001).