Turkish Journal of Mathematics
Abstract
Let $a,b$ be two commutative generalized Drazin invertible elements in a Banach algebra; the expressions for the generalized Drazin inverse of the product $ab$ and the sum $a+b$ were studied in some current literature on this subject. In this paper, we generalize these results under the weaker conditions $a^{2}b=aba$ and $b^{2}a=bab$. As an application of our results, we obtain some new representations for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra, extending some recent works.
DOI
10.3906/mat-1605-8
Keywords
Generalized Drazin inverse, Banach algebra, additive result, block matrix
First Page
548
Last Page
563
Recommended Citation
ZOU, H, MOSIC, D, & CHEN, J (2017). Generalized Drazin invertibility of the product and sum of two elements in a Banach algebra and its applications. Turkish Journal of Mathematics 41 (3): 548-563. https://doi.org/10.3906/mat-1605-8
