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

We consider infinite-dimensional generalized Hilbert matrices of the form $H_{i,j} = \frac{d_i d_j}{x_i + x_j}$, where $d_i$ are nonnegative weights and $x_i$ are pairwise disjoint positive numbers. We state sufficient and, for monotonically rearrangeable $x_i$, also necessary conditions for $d_i$, $x_i$ such that the induced operator from $\ell^2 \to \ell^2$ and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.