On operators whose core–EP inverse is n-potent

Mosić, Dijana and Zhang, Daochang and Hu, Jianping (2024) On operators whose core–EP inverse is n-potent. Miskolc Mathematical Notes, 25 (2). pp. 921-932. ISSN 1787-2413

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Abstract

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).

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