Moderate vs. severe ill-posedness for the composition T = H J

Determine whether the compact composition operator T = H J: L^2(0,1) → ℓ^2, where H is the Hausdorff moment operator [Hx]_j = ∫_0^1 x(t) t^{j-1} dt and J is the simple integration operator [Jx](s) = ∫_0^s x(t) dt, is moderately ill-posed (i.e., its singular values decay polynomially) or severely ill-posed (i.e., its singular values decay exponentially).

Background

The paper studies compositions of non-compact operators with non-closed range and how they affect the ill-posedness of the resulting operator. For T = H J, prior work established bounds on the singular values: there exist constants c̲, c̄ > 0 such that exp(−c̲ i) ≤ σ_i(T) ≤ c̄ i^−3/2 for sufficiently large i, which implies that the interval of ill-posedness lies within [3/2, ∞].

These bounds leave open whether the decay is ultimately polynomial (moderate ill-posedness) or exponential (severe ill-posedness), motivating a precise classification of the degree of ill-posedness for T = H J.