Exponential lower bound for singular values of T = D H (equivalently H C*)

Establish whether the compact operator T = D H = H C*: L^2(0,1) → ℓ^2, where H is the Hausdorff moment operator and D is the diagonal operator Dy = (y_j/j), admits a purely exponential lower bound on its singular values of the form σ_i(T) ≥ C exp(−c i) for some positive constants C and c, analogous to the lower bound known for T = H J.

Background

The paper proves an upper bound σ_i(T) ≤ c i−3/2 and a lower bound σ_i(T) ≥ (c_0/i) exp(−2 i) for T = D H = H C*. Numerical experiments suggest the proven lower bound may be pessimistic.

Motivated by these observations, the authors conjecture that a lower bound of purely exponential type, similar to the case T = H J, may hold for T = D H as well.

References

Note that the lower bound in Theorem~\ref{thm:lowerbounds} for $D \, H $ seem to be too pessimistic, and one might conjecture that a lower bound $\exp(-c i)$ as for $H \, J$ holds.

— Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces (2401.14701 - Kindermann et al., 26 Jan 2024) in Section: Numerical illustration of the decay of the singular values