Characterizations of the commutators and the anticommutator involving idempotents

In this paper, we characterize the idempotency, the invertibility or generalized invertibility, the positivity and the range relations of the commutator C=PQ-QP and the anticommutator D=PQ+QP involving idempotents. Particular attention is paid to the idempotent operators P and Q, and some further characteristics when P and Q are positive are derived as well.