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Proceedings of the American Mathematical Society

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Dominated estimates in Hilbert space


Authors: M. A. Akcoglu and H. D. B. Miller
Journal: Proc. Amer. Math. Soc. 55 (1976), 371-375
DOI: https://doi.org/10.1090/S0002-9939-1976-0397443-8
MathSciNet review: 0397443
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Abstract | References | Additional Information

Abstract: Let $ U$ be a unitary operator on a Hilbert space $ H$, and let $ {A_n}(U),n = 1,2, \ldots $, be the Cesàaro means of $ U$. It is shown that $ \Sigma _{n = 1}^\infty {P_n}{A_n}(U)$ is bounded for every sequence of mutually orthogonal projections $ {P_n},n = 1,2, \ldots $, if and only if $ 1$ is not a limit point of the spectrum of $ U$. The proof is obtained by adapting ideas of Menchoff and Burkholder to show that for any orthonormal sequence $ {f_n},n = 0, \pm 1, \pm 2, \ldots $, in $ H$, there is an orthonormal sequence $ {g_n},n = 1,2, \ldots $, such that

$\displaystyle \sum\limits_{k = 1}^n {\vert({f_1} + {f_2} + \cdots + {f_k},{g_k}){\vert^2} \geqslant \frac{1} {{36}}n{{(\log n)}^2}.} $


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

  • [1] M. A. Akcoglu and L. Sucheston, On the dominated ergodic theorem in $ {L_2}$ space, Proc. Amer. Math. Soc. 43 (1974), 379-382. MR 0333770 (48:12094)
  • [2] D. L. Burkholder, Semi-Gaussian subspaces, Trans. Amer. Math. Soc. 104 (1962), 123-131. MR 25 #2426. MR 0138986 (25:2426)
  • [3] D. Menchoff, Sur les séries de fonctions orthogonales, Fund. Math. 4 (1923), 82-105.
  • [4] G. Alexits, Konvergenzprobleme der orthogonalreihen, Akad. Kiado, Budapest, 1960; English transl., Internat. Series Monographs Pure and Appl. Math., vol. 20, Pergamon Press, New York, 1961. MR 28 #5292; 36 #1911 (pp. 88-99 contain an exposition of the results of [3]). MR 0162091 (28:5292)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0397443-8
Article copyright: © Copyright 1976 American Mathematical Society

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