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In this paper, we will introduce two new classes of generalized Drazin invertible operators on Hilbert space, which are called core–EP star and star core–EP operators, using the core–EP inverse and the adjoint of a given operator. We also represent here a few characterizations of these new operators from two points of view, algebraic and geometrical, and make relations to some familiar inverses, which are studied before. Next, we will consider two decompositions of Hilbert space and give the matrix representations of these new operators, following the decompositions. As the special section, there is the case when the operator is Drazin invertible. By using a *core–EP inverse, instead of the core–EP inverse, we get another new classes called *core–EP star and star *core–EP operators.

Core–EP Star and Star Core–EP Operators