The m-weak core inverse

Since the day the core inverse was known in a paper of Bakasarly and Trenkler, it has been widely researched. So far, there are four generalizations of this inverse for the case of matrices of an arbitrary index, namely, the BT inverse, the DMP inverse, the core-EP inverse and the WC inverse. In this paper we introduce a new type of generalized inverse for a matrix of an arbitrary index to be called m-weak core inverse which generalizes the core-EP inverse, the WC-inverse, and therefore the core inverse. We study several properties and characterizations of the m-weak core inverse by using matrix decompositions.