Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive Borel measure μ μ on ( 0 , 1 ) (0,1) we consider an infinite matrix Γ μ Γ _μ , related to the classical Hausdorff matrix defined by the same measure μ μ , in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ μ is the Lebesgue measure, Γ μ Γ _μ reduces to the classical Hilbert matrix. We prove that the matrices Γ μ Γ _μ are not Hankel, unless μ 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 > ∞ H^p, \, 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.