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Discrete Hausdorff Transformations

Gerald Leibowitz
Proceedings of the American Mathematical Society
Vol. 38, No. 3 (May, 1973), pp. 541-544
DOI: 10.2307/2038946
Stable URL: http://www.jstor.org/stable/2038946
Page Count: 4
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Discrete Hausdorff Transformations
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Abstract

Let K be a complex valued measurable function on (0, 1\rbrack such that $\int^1_0 t^{-1/p}| K(t)| dt$ is finite for some $p > 1$. Let H be the Hausdorff operator on lp determined by the moments μn = ∫1 0 tnK(t)dt. Define Ψ(z) = ∫1 0 tzK(t)dt. Then for each z with $\operatorname{Re} z > -1/p, \Psi(z)$ is an eigenvalue of H*. The spectrum of H is the union of $\{0 \}$ with the range of Ψ on the half-plane ℜ z ≥ -1/p.

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