The spectral properties of [m]-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 T-n is also an [m]-complex symmetric operator for any n is an element of N. In addition, we prove that if T is an [m]-complex symmetric operator, then sigma(a)(T), sigma(SVEP)(T), sigma(beta)(T), and sigma((beta)is an element of)(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.