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

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

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 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)****[4]**Ivan Niven and Herbert S. Zuckerman,*An introduction to the theory of numbers*, Wiley, New York, 1960. MR**0114786 (22:5605)**

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

Article copyright:
© Copyright 1992
American Mathematical Society