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.

中文翻译：

---

Moore-Penrose 逆的推广

为了将 Moore-Penrose 逆的符号从闭域算子扩展到广义 Drazin 可逆算子，我们提出了一个新的广义逆，称为广义 Moore-Penrose 逆。我们考虑了广义摩尔-彭罗斯逆的许多特征和不同的表示。受这些表示的启发，我们建立了最大类的算子，对于这些算子，广义 Moore-Penrose 逆的表示仍然有效。证明了广义摩尔-彭罗斯逆的一些规范形式。对偶广义摩尔-彭罗斯逆也被定义和研究。应用广义摩尔-彭罗斯和对偶广义摩尔-彭罗斯逆，我们求解一些线性方程组。