Complex sequences whose “moments” all vanish

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.