# GROUP INVERSE AND GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES IN A BANACH ALGEBRA

• Mosic, Dijana (Faculty of Sciences and Mathematics University of Nis)
• Published : 2014.05.31
• 146 5

#### Abstract

Necessary and sufficient conditions for the existence of the group inverse of an anti-triangular block matrix in Banach algebras are presented. Using these results, we give the formulae for the generalized Drazin inverse of a block matrix.

#### Keywords

generalized Drazin inverse;reverse order law;block matrix

#### References

1. N. Castro-Gonzalez and J. J. Koliha, New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 6, 1085-1097. https://doi.org/10.1017/S0308210500003632
2. C. Bu, J. Zhao, and K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra 18 (2009), 117-125.
3. S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979.
4. N. Castro-Gonzalez and E. Dopazo, Representations of the Drazin inverse for a class of block matrices, Linear Algebra Appl. 400 (2005), 253-269. https://doi.org/10.1016/j.laa.2004.12.027
5. C. Cao and J. Li A note on the group inverse of some $2{\times}2$ block matrices over skew fields, Appl. Math. Comput. 217 (2011), no. 24, 10271-10277. https://doi.org/10.1016/j.amc.2011.05.027
6. C. Deng and Y. Wei, A note on the Drazin inverse of an anti-triangular matrix, Linear Algebra Appl. 431 (2009), no. 10, 1910-1922. https://doi.org/10.1016/j.laa.2009.06.030
7. D. S. Djordjevic and Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (2002), no. 1, 115-125. https://doi.org/10.1017/S1446788700008508
8. R. E. Hartwig, G. Wang, and Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001), no. 1-3, 207-217. https://doi.org/10.1016/S0024-3795(00)00257-3
9. J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), no. 3, 367-381. https://doi.org/10.1017/S0017089500031803
10. X. Liu and H. Yang, Further results on the group inverses and Drazin inverses of antitriangular block matrices, Appl. Math. Comput. 218 (2012), no. 17, 8978-8986. https://doi.org/10.1016/j.amc.2012.02.058

#### Cited by

1. Representations for generalized Drazin inverse of operator matrices over a Banach space vol.40, pp.13036149, 2016, https://doi.org/10.3906/mat-1502-27