Generalized Hilbert operators arising from Hausdorff matrices

IF 0.8 3区数学 Q2 MATHEMATICS

Proceedings of the American Mathematical Society Pub Date : 2024-05-11 DOI:10.1090/proc/16917

C. Bellavita, N. Chalmoukis, V. Daskalogiannis, G. Stylogiannis

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When <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Lebesgue measure, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript mu\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\Gamma _\\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces to the classical Hilbert matrix. 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引用次数: 0

Abstract

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

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豪斯多夫矩阵产生的广义希尔伯特算子

对于 ( 0 , 1 ) (0 , 1) 上的有限正伯尔量μ \mu，我们考虑一个无穷矩阵Γ μ \Gamma _\mu ，它与由相同量μ \mu 定义的经典豪斯多夫矩阵相关，其代数方式与希尔伯特矩阵与塞萨罗矩阵相关的代数方式相同。当 μ \mu 是 Lebesgue 度量时，Γ μ \Gamma _\mu 等同于经典的希尔伯特矩阵。我们证明了矩阵 Γ μ Gamma _\mu 不是汉克尔矩阵，除非 μ \mu 是 Lebesgue 量的常数倍，我们给出了它们在哈代空间 H p , 1 ≤ p > ∞ H^p, \, 1 \leq p > \infty 的尺度上有界的必要条件和充分条件，并研究了它们的紧凑性和完全连续性。在 2 ≤ p > ∞ 2\leq p>\infty 的情况下，我们能够计算出算子的精确规范值。

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来源期刊

Proceedings of the American Mathematical Society 数学-数学

CiteScore

1.70

自引率

10.00%

发文量

207

审稿时长

2-4 weeks

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