A criterion for the \(\mathbb Z_d\)-symmetry of the spectrum of a compact operator (Q2835248)

Let \(T: X\to X\) be a linear operator in a finite-dimensional space \(X\). It is easy to see that the following conditions are equivalent: (a) the spectrum of \(T\) is symmetric, or \(\mathbb Z_2\)-symmetric, i.e., \(\lambda\in\sigma (T)\Rightarrow -\lambda\in\sigma (T)\) and their algebraic multiplicities \(m(\lambda), m(-\lambda)\) are equal; (b) the trace \(\operatorname {tr}\left(T^p\right)=0\) for all odd \(p\in \mathbb N\). In [\textit{M. Zelikin}, Dokl. Akad. Nauk 418, 737--740 (2008; Zbl 1166.47008)] it was proved that this claim could be extended to the trace-class operators in a Hilbert space. In this paper, it is proved that such claims could be made (i) for the \(\mathbb Z_d\)-symmetry of a spectrum, \(d\geq2\); (ii) in general Banach spaces.