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

TL;DR

This paper establishes a criterion for the $bZ_d$--symmetry of the spectrum of a compact operator in a Banach space, based on the vanishing of certain traces of its powers, especially when some power is nuclear.

Contribution

It provides a new spectral symmetry criterion for compact operators with nuclear powers, linking trace conditions to $bZ_d$--symmetry.

Findings

Spectral symmetry characterized by trace vanishing conditions.

Criterion applies when some power of the operator is nuclear.

Results extend understanding of spectral properties of compact operators.

Abstract

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.