The representations of the g-Drazin inverse in a Banach algebra

Year 2021, Volume: 50 Issue: 3, 659 - 667, 07.06.2021

Marjan Sheibani Abdolyousefi

https://doi.org/10.15672/hujms.754006

Cited By: 4

Abstract

The aim of this paper is to establish an explicit representation of the generalized Drazin inverse $(a+b)^d$ under the condition $$ab^2=0, ba^2=0, a^{\pi}b^{\pi}(ba)^2=0.$$ Furthermore, we apply our results to give some representation of generalized Drazin inverse for a $2\times 2$ block operator matrix. These extend the results on Drazin inverse of Bu, Feng and Bai [Appl. Math. Comput. 218, 10226-10237, 2012] and Dopazo and Martinez-Serano [Linear Algebra Appl. 432, 1896-1904, 2010].

Keywords

g-Drazin inverse, additive property, perturbation, Banach algebra

References

• [1] J. Benítez, X. Qin and X. Liu, New additive results for the generalized Drazin inverse in a Banach Algebra, Filomat 30 (8), 2289-2294, 2016.
• [2] C. Bu, C. Feng and S. Bai, Representations for the Drazin inverses of the sum of two matrices and some block matrices, Appl. Math. Comput. 218, 10226-10237, 2012.
• [3] S. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear Multilinear Algebra 14, 195-198, 1983.
• [4] S.L. Campbell and C.D. Meyer, Generalized inverses of linear transformations, SIAM, 2009.
• [5] N. Castro-González and J.J. Koliha, New additive results for the G- Drazin inverse, Proc. Roy. Soc. Edinburgh. Sect A, 134, 1084-1097, 2004.
• [6] C. Deng, D.S. Cvetcović-Ilić and Y. Wei, Some results on the genrealized Derazin inverse of operator matrices, Linear Multilinear Algebra 58, 503-521, 2010.
• [7] E. Dopazo and M.F. Martinez-Serrano, Further results on the representation of the Drazin inverse of a $2\times2$ block matrix, Linear Algebra Appl., 432, 1896-1904, 2010.
• [8] Y. Liao, J. Chen and J. Cui, Cline’s formula for the generalized Drazin inverse, Bull. Malays. Math. Sci. Soc. 37, 37–42, 2014.
• [9] D. Mosić and D.S. Djordjević, Block representations of the generalized Drazin inverse, Appl. Math. Comput. 331, 200-209, 2018.
• [10] H. Yang and X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Applied Math. 235, 1412-1417, 2011.
• [11] X. Zhang and G. Chen, The computation of Drazin inverse and its applications in Markov chains, Appl. Math. Comput. 183, 292-300, 2006.