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Proceedings of the American Mathematical Society

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



Dominated estimates in Hilbert space

Authors: M. A. Akcoglu and H. D. B. Miller
Journal: Proc. Amer. Math. Soc. 55 (1976), 371-375
MathSciNet review: 0397443
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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}.} $

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Article copyright: © Copyright 1976 American Mathematical Society