Generalized Hilbert operators arising from Hausdorff matrices. (English) Zbl 07928595
Summary: 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 Cesáro 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 \(H^p\), \(1\leq p<\infty\), and we study their compactness and complete continuity properties. In the case \(2\leq p<\infty\), we are able to compute the exact value of the norm of the operator.
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