Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive Borel measure μ on (0, 1) (0, 1) we consider an infinite matrix Γ μ _, related to the classical Hausdorff matrix defined by the same measure μ, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ is the Lebesgue measure, Γ μ _ reduces to the classical Hilbert matrix. We prove that the matrices Γ μ _ are not Hankel, unless μ 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 > ∞ Hᵖ, \, 1 p >, and we study their compactness and complete continuity properties. In the case 2 ≤ p > ∞ 2 p>, we are able to compute the exact value of the norm of the operator.