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We define the class of (C p ,α)-hyponormal operators and study the inclusion between such classes under various hypotheses for p and α, and then obtain some sufficient conditions for the self-commutator of the Aluthge transform T = |T| 1 2 U|T| 2 of (Cp, α)-hyponormal operators to be in the trace-class and have trace zero.

$(\mathcal {C}_{p}, \alpha )$-hyponormal operators and trace-class self-commutators with trace zero