On s ‐numbers and Weyl inequalities of operators in Banach spaces

Let s = ( s n ) 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( Π i = 1 n | λ i ( T ) | )1 / n ⩽ c ( δ ) ω C 2 ( X ) ( 1 + ln ω C 2 ( X ) ) ( Π i = 1n δS i ( T ) ) 1 / n δ, where wC 2 ( X ) is the weak cotype 2 constant of X , n δ ≔ [ n /(1+δ)] and c (δ) ⩽ c 0 (1+1/δ ln (1/δ)) with an absolute constant c 0 ⩾ 1.