Generalized Hilbert operators arising from Hausdorff matrices

2023/07/28 by Bellavita, Carlo, Chalmoukis, Nikolaos, Daskalogiannis, Vassilis +1 · 1 citation
#30H10 #47B91 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2307.15334

Abstract

For a finite, positive, Borel measure μ on (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 < ∞, and we study their compactness and complete continuity properties. In the case 2≤ p<∞, we are able to compute the exact value of the norm of the operator.

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