Extensions of generalized core-EP inverse

The intention of our research is to define and investigate some extensions of generalized core-EP (or shortly GCEP) inverse. Inspired by Urquhart expression for the outer inverses, we firstly introduce Φ _1 -GCEP inverse replacing A^(2)_ℛ(B),𝒩(C) in the definition of the GCEP inverse by the more general expression Φ _1:=B(CAB)^(1)C . Further, Φ _2 -GCEP inverse is defined stating Φ _2:=B(CAB)^(2)C instead of A^(2)_ℛ(B),𝒩(C) in GCEP inverse. The most comprehensive form of the GCEP inverse is presented as Φ -GCEP inverse based on the replacement A^(2)_ℛ(B),𝒩(C) with arbitrary matrix Φ with an adequate size. Main properties, representations and characterizations of Φ _1 -GCEP inverse, Φ _2 -GCEP inverse and Φ -GCEP inverse are established as well as algorithms for their calculating. Solvability of some linear matrix equations are investigated applying these three new outer inverses.