Generalization of the Moore-Penrose inverse

In order to extend the notation of the Moore-Penrose inverse from an operator with closed range to a generalized Drazin invertible operator, we present a new generalized inverse which is called the generalized Moore-Penrose inverse. We consider a number of characterizations and different representations of the generalized Moore-Penrose inverse. Inspired by these representations, we establish maximal classes of operators for which the representations of the generalized Moore-Penrose inverse are still valid. Some canonical forms for the generalized Moore-Penrose inverse are proved. The dual generalized Moore-Penrose inverse is defined and investigated too. Applying the generalized Moore-Penrose and dual generalized Moore-Penrose inverses, we solve some systems of linear equations.