GENERALIZED HILBERT OPERATORS ARISING FROM HAUSDORFF MATRICES

For a finite, positive Borel measure mu on (0,1) we consider an infinite matrix Gamma(mu), related to the classical Hausdorff matrix defined by the same measure mu, in the same algebraic way that the Hilbert matrix is related to the Ces & aacute;ro matrix. When mu is the Lebesgue measure, Gamma(mu) reduces to the classical Hilbert matrix. We prove that the matrices Gamma(mu) are not Hankel, unless mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H-p, 1 <= p < infinity, and we study their compactness and complete continuity properties. In the case 2 <= p < infinity, we are able to compute the exact value of the norm of the operator.