Discrete Hausdorff transformations
HTML articles powered by AMS MathViewer
- by Gerald Leibowitz
- Proc. Amer. Math. Soc. 38 (1973), 541-544
- DOI: https://doi.org/10.1090/S0002-9939-1973-0315508-0
- PDF | Request permission
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 $\operatorname {Re} z \geqq - 1/p$.References
- Richard Arens and I. M. Singer, Generalized analytic functions, Trans. Amer. Math. Soc. 81 (1956), 379–393. MR 78657, DOI 10.1090/S0002-9947-1956-0078657-5
- G. H. Hardy, An inequality for Hausdorff means, J. London Math. Soc. 18 (1943), 46–50. MR 8854, DOI 10.1112/jlms/s1-18.1.46
- Gerald Leibowitz, Spectra of discrete Cesàro operators, Tamkang J. Math. 3 (1972), 123–132. MR 318948
- B. E. Rhoades, Spectra of some Hausdorff operators, Acta Sci. Math. (Szeged) 32 (1971), 91–100. MR 305108
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- 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