ON OPERATORS WHOSE CORE-EP INVERSE IS n-POTENT

The main contribution of this paper is to establish a number of equivalent conditions for the core-EP inverse of an operator, to be n-potent. We prove that the core-EP inverse of an operator is n-potent if and only if the Drazin inverse of the same operator is n-potent. Thus, we present new characterizations for n-potency of the Drazin inverse. Consequently, we get many characterizations for the core-EP inverse (and Drazin inverse) to be an idempotent. We observe that the core-EP inverse of an operator is idempotent if and only it is the orthogonal projector. Furthermore, we show that the n-potency of an operator implies n-potency of its core-EP inverse and develop the condition for the converse to hold. Applying these results, we obtain necessary and sufficient conditions for the n-potency and idempotency of the core inverse. Notice that the core inverse of an operator is n-potent (or idempotent) if and only if the given operator is n-potent (idempotent).