Reverse order laws for the Drazin inverses

Reverse-order law for the Moore–Penrose inverses of tensors

“…The study of this problem for generalized inverses in C*-algebras can be seen in the work by Cvetkovic-Ilic and Hartee [7] and Mosic and Djordjevic [17]. While Deng [8] studied the same problem for the group invertible operators, Wang et al [22] considered for the Drazin invertible operators. The vast work on the reverse order law and its several multivariety extensions in different areas of mathematics in the literature and the recent works in [19] and [3] motivate us to study this problem in the framework of tensors.…”

Further results on the Drazin inverse of even‐order tensors

“…The Drazin inverse of a tensor via Einstein product plays significant roles in singular linear tensor equation 14 . In this section, we further embellish on this theory by producing a few more characterizations of this inverse, which is a generalization of the Drazin inverse of matrices 3,13,40‐43 . This section is divided into two parts.…”

Reverse order law of Drazin inverse for bounded linear operators

“…Therefore, Proof. Since N 14 and Q 44 are nilpotent, it is easy to check that N 14 + Q 44 is also nilpotent by (14). Then, the conclusion can be obtained according to the proof of Theorem 2.6.…”

“…Obviously, (PQ) D Q D P D . Thus, many authors were attracted to investigate the condition under which reverse order laws hold for various generalized inverses and some interesting results were obtained [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

“…For bounded linear operators, Deng [4,14] presented necessary and sufficient conditions for the reverse order law of group inverse and Drazin inverse, respectively. Mosić [8,10] investigated the reverse order law for the generalized Drazin inverse in Banach algebras.…”