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=didjxi+xj, where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, for monotonically rearrangeable xi, also necessary conditions for di, xi such that the induced operator from l2→l2 and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.