On a new generalized inverse for Hilbert space operators

Using the Moore-Penrose inverse and the core-EP inverse, we define a new generalized inverse (called the MPCEP inverse) for a Hilbert space operator. Several equivalent conditions for a Hilbert space operator to be the MPCEP inverse are presented. We also give a canonical form for the MPCEP inverse. Various kinds of generalized inverses are integrated by MPCEP inverse and many equivalent relations among these inverses are investigated using appropriate idempotents. Also, we develop maximal classes of operators for which the representation of the MPCEP inverse is still valid. We introduce and study a new binary relation associated with the MPCEP inverse.