The spectral properties of [m] $[m]$-complex symmetric operators

Abstract

Abstract In this paper, we introduce the class of [m]-complex symmetric operators and study various properties of this class. In particular, we show that if T is an [m]-complex symmetric operator, then Tn $T^{n}$ is also an [m]-complex symmetric operator for any n∈N $n\in\mathbb {N}$. In addition, we prove that if T is an [m]-complex symmetric operator, then σa(T) $\sigma_{a}(T)$, σSVEP(T) $\sigma_{\mathrm{SVEP}}(T)$, σβ(T) $\sigma_{\beta }(T)$, and σ(β)ϵ(T) $\sigma_{(\beta)_{\epsilon}}(T)$ are symmetric about the real axis. Finally, we investigate the stability of an [m]-complex symmetric operator under perturbation by nilpotent operators commuting with T.

Keywords

• [ m ] $[m]$ -complex symmetric operator
• Property (β)
• Nilpotent operator