Criterion for mathbb{Z}_d--symmetry of a Spectrum of a Compact Operator

If $A$ is a compact operator in a Banach space and some power $A^q$ is nuclear we give a criterion of $\mathbb{Z}_{d}$ -- symmetry of its spectrum $\sigma{A}$ in terms of vanishing of the traces $\mathop{\mathit{Trace}} A^n$ for all $n$, $n \geq 0$, $n \neq 0 \mod d$, sufficiently large.