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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Discrete Hausdorff transformations


Author: Gerald Leibowitz
Journal: Proc. Amer. Math. Soc. 38 (1973), 541-544
MSC: Primary 47B99; Secondary 40H05, 47A10
DOI: https://doi.org/10.1090/S0002-9939-1973-0315508-0
MathSciNet review: 0315508
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Abstract: Let $ K$ be a complex valued measurable function on $ (0,1]$ such that $ \int_0^1 {{t^{ - 1/p}}\vert K(t)\vert dt} $ is finite for some $ p > 1$. Let $ H$ be the Hausdorff operator on $ {l^p}$ determined by the moments $ {\mu _n} = \int_0^1 {{t^n}K(t)} dt$. Define $ \Psi (z) = \int_0^1 {{t^z}K(t)} dt$. Then for each $ z$ with Re $ \operatorname{Re} z > - 1/p,\Psi (z)$ is an eigenvalue of $ {H^\ast}$. The spectrum of $ H$ is the union of $ \{ 0\} $ with the range of $ \Psi $ on the half-plane Re $ \operatorname{Re} z \geqq - 1/p$.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1973-0315508-0
Article copyright: © Copyright 1973 American Mathematical Society

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