We prove the existence of a nontrivial singular trace τ defined on an ideal J closed with respect to the logarithmic submajorization such that τ(Aρ(α)) = 0, where Aρ(α) ∶ L 2(0, 1) → L 2(0, 1), [Aρ(α) f](θ) = ∫ 1 0 ρ(αθ/x) f (x)dx , 0 < α ≤ 1. We also show that τ(Aρ(α)) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr (Ar ρ (α)), r ∈ N, r ≥ 2.
- SINGULAR TRACE
- SPECTRAL TRACE
- modified Fredholm determinant
- Riemann hypothesis