Generalized Drazin invertibility of combinations of idempotents

Abstract

The paper serves as a correction to J. Math. Anal. Appl. 359 (2009) 731-738 dealing with the Drazin invertibility of combinations of idempotents p, q in a Banach algebra. As the arguments employed to do this work equally well for both the usual and generalized Drazin inverse, the latter is included in the discussion which covers the equivalence of the Drazin invertibility of 1 − pq, p − q, p + q and in general of αp + βq with αβ = 0, as well as the equivalence of the Drazin invertibility of the commutator pq − qp and anticommutator pq + qp of p, q.

Idempotents generated by weighted generalized inverses in rings with involution

“…(i) a is EP. (ii) a ∈ R (1,3) , aR = a * R. (iii) a ∈ R (1,4) , aR = a * R. (iv) a ∈ R # , a k = a k+1 a (1,3) for any positive integer k. (v) a ∈ R # , a k = a (1,4) a k+1 for any positive integer k.…”

“…It is well known that idempotents are a class of important elements and has a close relationship with generalized inverses. Many researchers have considered questions concerning the idempotents in various fields, such as in complex matrices, Banach algebras, rings, etc (see [3], [4], [6], [7], [9], [10]). Therein, characterizations of idempotents generated by Moore-Penrose inverses and weighted Moore-Penrose inverses of elements over various sets attract wide interest from many scholars.…”

“…Shortly after, Mosić and Djordjević [9] derived the characterization of aa † e, f = bb † e, f , when a, b ∈ R † e, f . Recently, Zhu and Peng [14] gave several characterizations of a (1,4) a = bb (1,3) for a ∈ R (1,4) and b ∈ R (1,3) . In this paper, we mainly consider properties and characterizations of idempotents generated by weighted Moore-Penrose generalized inverses and weighted pseudo core inverses, respectively.…”

Drazin invertibility of product and difference of idempotents in a ring

“…Then, (1) p − q ∈ A D if and only if p + q ∈ A D if and only if 1 − pq ∈ A D ; (2) pq − qp ∈ A D if and only if pq + qp ∈ A D if and only if pq ∈ A D and p − q ∈ A D . These results are generalizedto Banach algebra in [10]. More results on the Drazin invertibility of sum, difference and product of idempotents can be found in [3-8, 10, 13].In this paper, we consider the Drazin invertibility of p − q, pq, pq − qp (commutator) and pq + qp (anti-commutator), where p and q are idempotents in a ring.…”

“…to Banach algebra in [10]. More results on the Drazin invertibility of sum, difference and product of idempotents can be found in [3-8, 10, 13].…”