An extension of Cline''s formula for a generalized Drazin inverse

Abstract

In this note we give an answer to a question recently posed by Zeng and Zhong, to note that Cline's formula for a generalized Drazin inverse extends to the case when aba = aca . Cline's formula for a pseudo Drazin inverse is also presented in this case.

New extensions of Cline’s formula for generalized inverses

“…An operator T ∈ B(X) is called B-Fredholm if for some n ∈ N the range R(T n ) is closed and the restriction T| R(T n ) of T to R(T n ) is Fredholm. In this case, [ We conclude this paper by an example to illustrate that the results obtained in this paper are proper generalizations of the corresponding ones in [18,33]. Example 2.9.…”

“…0, but ACD = DBD = Drazin invertibility) of AC and CD can only be deduced directly from the results obtained in this paper, and not from the corresponding ones in [18,33].…”

“…Their proof are given through the bridges quasipolar and pseudopolar, respectively. Very recently, the authors [18,33]…”

Cline’s formula for g-Drazin inverses in a ring

“…For any elements a, b in a ring R, it is well known that ab ∈ R qnil if and only if ba ∈ R qnil (see [7,Lemma 2.2]). We now generalized this fact as follows.…”

On then-strong Drazin invertibility in rings

“…This formula is called Cline's formula for the Drazin inverse. Many researchers considered Cline's formula for various types of generalized inverses, such as (b, c)-inverse [10], Mary inverse [27], Hirano inverse [2], pseudo-Drazin inverse [20], generalized Drazin inverse [13,14,16,23,24]. In [23], Zeng et al extended Cline's formula for the (pseudo, generalized) Drazin inverse to more general case.…”