Characterizations of the commutators and the anticommutator involving idempotents

摘要

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.