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Proceedings of the American Mathematical Society

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



Complex sequences whose ``moments'' all vanish

Author: W. M. Priestley
Journal: Proc. Amer. Math. Soc. 116 (1992), 437-444
MSC: Primary 40A99; Secondary 30D10, 47B15
MathSciNet review: 1097350
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Abstract: Must a sequence $ \{ {z_k}\} $ of complex numbers be identically zero if $ \sum {f({z_k}) = 0} $ for every entire function $ f$ vanishing at the origin? Lenard's example of a nonzero sequence of complex numbers whose power sums ("moments") all vanish is shown to give a negative answer to this question and to lead to a novel representation theorem for entire functions.

On the positive side it is proved that if $ \{ {z_k}\} $ is in $ {l^p}$ where $ p < \infty $, then vanishing moments imply $ \{ {z_k}\} $ is identically zero. Virtually the same proof shows that, on a Hubert space, two compact normal operators $ A$ and $ B$ with trivial kernels are unitarily equivalent if some power of each belongs to the trace class and $ \operatorname{tr}({A^n}) = \operatorname{tr}(B^n)$ for all $ n$ in a set of positive integers with asymptotic density one.

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

  • [1] Don Deckard, Complete sets of unitary invariants for compact and trace-class operators, Acta Sei. Math. (Szeged) 28 (1967), 9-20. MR 0217643 (36:732)
  • [2] Don Deckard and Carl Pearcy, On unitary invariants of Hilbert-Schmidt operators, Proc. Amer. Math. Soc. 16 (1965), 671-675. MR 0179620 (31:3866)
  • [3] Andrew Lenard, A nonzero complex sequence with vanishing power-sums, Proc. Amer. Math. Soc. 108 (1990), 951-953. MR 1009993 (90i:40007)
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Article copyright: © Copyright 1992 American Mathematical Society

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