On the type of ill-posedness of generalized Hilbert matrices and related operators

We consider infinite-dimensional generalized Hilbert matrices of the form Hi,j = dᵢ dⱼ/xᵢ + xⱼ, where dᵢ are nonnegative weights and xᵢ are pairwise disjoint positive numbers. We state sufficient and, for monotonically rearrangeable xᵢ, also necessary conditions for dᵢ, xᵢ such that the induced operator from ² → ² and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.