On the Drazin Inverse of the Sum of Two Matrices

Abstract

We deduce the explicit expressions for(P+Q)Dand(PQ)Dof two matricesPandQunder the conditionsP2Q=PQPandQ2P=QPQ. Also, we give the upper bound of|P+QD-PD|2.

Additive and Product Properties of Drazin Inverses of Elements in a Ring

“…Moreover, the expressions of (ab) D and (a + b) D are presented. Consequently, some results in [9,12,16] can be deduced from our results.…”

“…Under the conditions P 2 Q = P QP and Q 2 P = QP Q, Liu, Wu and Yu [9] characterized the relations between the Drazin inverses of P + Q and 1 + P D Q for complex matrices P and Q by using the methods of splitting complex matrices into blocks. In this paper, we extend the results in [9] to a ring R. For a, b ∈ R D , it is shown that ab ∈ R D and that a + b ∈ R D if and only if 1 + a D b ∈ R D under the conditions a 2 b = aba and b 2 a = bab. Moreover, the expressions of (ab) D and (a + b) D are presented.…”

Generalized Drazin invertibility of the product and sum of two elements in a Banach algebra and its applications

“…However, ab ̸ = ba. The next result was proved for complex matrices (see [13,Lemma 2.3]). Indeed, it is true in a Banach algebra.…”

“…In [13], Liu et al deduced the explicit expressions for the Drazin inverses of the product ab and the sum a + b under the conditions a 2 b = aba and b 2 a = bab, where a and b are complex matrices. In [18], the corresponding results of [13] were studied for the pseudo Drazin inverse (which is a special case of generalized Drazin inverse [17]) in a Banach algebra. In this paper, we will further consider the results of [13] and [18] (1) S is invertible, A π BC = 0, CA π B = 0 , and AA π B = A π BD (see [5]);…”

Perturbation for group inverse in a Banach algebra

“…The generalized Drazin inverse of a + b in a Banach algebra was studied in [5, Theorem 2.1] under the condition ba 2 = 0 and b 2 = 0. In [8], Liu et al investigated the Drazin inverse (P + Q) D of two complex matrices P and Q under the conditions P 2 Q = P QP and Q 2 P = QP Q. In [13], Zou et al presented the generalized Drazin inverse of a + b in a Banach algebra under a 2 b = aba and b 2 a = bab.…”