Some additive results for the generalized Drazin inverse in a Banach algebra

Abstract

Abstract. In this note, additive results are presented for the generalized Drazin inverse in Banach algebra. Necessary and sufficient conditions are given for the generalized Drazin invertibility of the sum of two commuting generalized Drazin invertible elements. These results are a generalization of the results from the paper [C.Y. Deng and Y. Wei. New additive results for the generalized Drazin inverse. J. Math. Anal. Appl., 370:313-321, 2010.] to the Banach algebra case.

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

“…Hence b ‡ = (1 − e)x ‡ . Using (i), we see x ‡ is given by (7) and (8). Following an analogous strategy as in the proof for y of (i), we have (ii) for y.…”

“…In 2010, Deng and Wei [8] derived a result under the condition PQ = QP, where P, Q are bounded linear operators. In 2011, Cvetković-Ilić, Liu and Wei [7] extended the result of [8] to Banach algebras. In 2014, Zhu, Chen and Patrício [19] obtained a result about the p-Drazin inverse of a + b under the conditions a 2 b = aba and b 2 a = bab which are weaker than ab = ba in Banach algebras.…”

“…In 2014, Zhu, Chen and Patrício [19] obtained a result about the p-Drazin inverse of a + b under the conditions a 2 b = aba and b 2 a = bab which are weaker than ab = ba in Banach algebras. More results on (generalized, pseudo) Drazin inverse can be found in [1,5,7,9,15,18].…”

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

“…Moreover, the expression of (P + Q) d was given. In [3], Cvetković-Ilić et al extended the result of [6] to Banach algebras.…”

Additive property of pseudo Drazin inverse of elements in Banach algebras

“…If ind(a) = 1, then b is the group inverse of a and is denoted by a # . In 2012, Wang and Chen [16] introduced the notion of pseudo More results on (generalized) Drazin inverse can be found in [1][2][3][4][5][6]8,9,[11][12][13][14][15].…”