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Abstract??Let (sn) be ans-number sequence. We show for eachk= 1, 2, . . . and n ?k+ 1 the inequalitybetween the eigenvalues ands-numbers of a compact operatorTin a Banach space. Furthermore, the constant (k+ 1)1/2is optimal forn=k+ 1 andk= 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicatives-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues. [ABSTRACT FROM AUTHOR]

Let s = (sn) be an injective s-number sequence in the sense of Pietsch. We show the following Weyl inequality between geometric means of eigenvalues and s-numbers for a Riesz-operator T: X → X acting on a (complex) Banach space of weak type 2: for any 0 < δ ≤ 1 and all n ∈ ℕ, we have , where wC2(X) is the weak cotype 2 constant of X, nδ ≔ [n/(1+δ)] and c(δ) ≤ c0 (1+1/δ ln (1/δ)) with an absolute constant c0 ≥ 1. [ABSTRACT FROM AUTHOR]