Generalized core-EP inverse for square matrices

We introduce, investigate, and apply two types of generalized inverses for square complex matrices using an arbitrary inner inverse of the input matrix and its core-EP inverse. Introduced combinations of inner inverses and the core-EP inverse are termed inner core-EP (ICEP) and core-EP inner (CEPI) inverse. Additionally, we extend the notion of the P-core inverse for square matrices with arbitrary indices. A few equivalent characterizations and effective representations of the introduced generalized inverses are suggested. In addition, representations of these inverses are established via core-EP and HS decompositions. The induced binary relations for these inverses are introduced and their properties are considered. An application of the proposed inverses in solving linear systems is also discussed.