# Journal of Operator Theory

Volume 76, Issue 1, Summer 2016 pp. 57-65.

A criterion for the $\mathbb{Z}_d$-symmetry of the spectrum of a compact operator

**Authors**:
Boris S. Mityagin

**Author institution:** Department of Mathematics, The Ohio State University, Columbus, 43201, U.S.A.

**Summary: ** 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 $\mathrm{Tr}A^n$ for all $n$, $n \geqslant 0$, $n \neq 0 \mod d$, sufficiently large.

**DOI: **http://dx.doi.org/10.7900/jot.2015jul24.2077

**Keywords: ** nuclear operators, trace, Cauchy--Riesz formula

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