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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 $\int _0^1 {{t^{ - 1/p}}|K(t)|dt}$ 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 $\operatorname {Re} z \geqq - 1/p$.


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Article copyright: © Copyright 1973 American Mathematical Society