The spectral properties of [Formula: see text] -complex symmetric operators

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 [Formula: see text] is also an [m]-complex symmetric operator for any [Formula: see text] . In addition, we prove that if T is an [m]-complex symmetric operator, then [Formula: see text] , [Formula: see text] , [Formula: see text] , and [Formula: see text] 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.