On generalized-Drazin inverses and GD-star matrices

Motivated by the works of Wang and Liu (Linear Algebra Appl 488:235–248, 2016) and Mosić (Results Math 75(2):1–21, 2020), we provide further results on GD inverses, and then introduce two new classes for square matrices called GD-star (generalized-Drazin-star) and GD-star-one (generalized-Drazin-star-one) using a GD inverse of a matrix. We exploit their various properties and characterize them in terms of various generalized inverses. We present a representation of a GD-star matrix by using the core-EP decomposition and Hartwig–Spindelböck decomposition. We also define a binary relation called GD-star order using this class of matrices. Further, we obtain some analogous results for the class of star-GD matrices. Moreover, the reverse-order law and forward-order law for GD inverse along with its monotonicity criteria are obtained.