A note on Cline’s formula for the generalized Drazin inverse

Abstract

Liao et al. proved that if the product ab is generalized Drazin invertible, then so is ba extending the Cline's formula to the case of the generalized Drazin invertibility. In this paper, we show that if (ab) n+1 is generalized Drazin invertible for arbitrary n ∈ N, then (ba) n is generalized Drazin invertible in a Banach algebra. So, we generalize Cline's formula to the case of the generalized Drazin invertibility of the powers of the products of elements a and b. Also, we prove that (ab) n+1 is generalized Drazin invertible if and only if ab is generalized Drazin invertible; and a n+1 is generalized Drazin invertible if and only if a is generalized Drazin invertible.

On Jacobson's lemma and Cline's formula for Drazin inverses

“…We remark that, for b = c in the previous results, we recover some results from [4,10,20], and, for b = c and a = d, results from [2,3,11,13,15,19,22]. Also, in these cases, we can get expressions for the corresponding inverses.…”

“…In 1965, Cline [3] showed that if ab is Drazin invertible, then ba is Drazin invertible too, and the so-called Cline's formula (ba) D = b((ab) D ) 2 a holds. In [11,13,19], Cline's formula was generalized to the case of generalized Drazin invertibility.…”

On the pseudo drazin inverse of the sum of two elements in a Banach algebra

“…Hence a ∈ A pD and a ‡ A = y. The next result is well-known for the Drazin inverse and the generalized Drazin inverse [14], and it is equally true for the p-Drazin inverse. Lemma 2.8.…”

Characterizations of annihilator $(b,c)$-inverses in arbitrary rings

“…In this section, we generalize Cline's formula to the case of annihilator (b, c)-inverses. The ideas arose most directly from [3, Section 6] and [22,Theorem 2.1].…”