In this paper, we study the structure of Δ m(T) defined by the following: Δm(T):=∑j=0m(-1)m-j(mj)T*jCTm-jC. In particular, we prove that if m is even, then Δ m(T) is complex symmetric with the conjugation C, and if m is odd, then Δ m(T) is skew complex symmetric with the conjugation C. Moreover, we investigate the conditions for (m+ 1)-complex symmetric operators to be m-complex symmetric operators and characterize the spectrum of Δ m(T). Finally, we show that if T∈ L(H) is Hermitian or Δ 1(T) is p-hyponormal, then Δ 2(T) = 0 implies Δ 1(T) = 0.
- Berberian’s method
- m-complex symmetric operator