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 of complex numbers be identically zero if for every entire function 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 is in where , then vanishing moments imply is identically zero. Virtually the same proof shows that, on a Hubert space, two compact normal operators and with trivial kernels are unitarily equivalent if some power of each belongs to the trace class and for all in a set of positive integers with asymptotic density one.

**[1]**Don Deckard,*Complete sets of unitary invariants for compact and trace-class operators*, Acta Sci. Math. (Szeged)**28**(1967), 9–20. MR**0217643****[2]**Don Deckard and Carl Pearcy,*On unitary equivalence of Hilbert-Schmidt operators*, Proc. Amer. Math. Soc.**16**(1965), 671–675. MR**0179620**, 10.1090/S0002-9939-1965-0179620-7**[3]**Andrew Lenard,*A nonzero complex sequence with vanishing power-sums*, Proc. Amer. Math. Soc.**108**(1990), no. 4, 951–953. MR**1009993**, 10.1090/S0002-9939-1990-1009993-9**[4]**Ivan Niven and Herbert S. Zuckerman,*An introduction to the theory of numbers*, John Wiley & Sons, Inc., New York-London, 1960. MR**0114786**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1097350-0

Article copyright:
© Copyright 1992
American Mathematical Society