  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.
