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## The generalized Drazin inverse of operator matrices

#### Li GUO [1] , Honglin ZOU [2] , Jianlong CHEN [3]

Representations for the generalized Drazin inverse of an operator matrix $\begin{pmatrix}A & B \\ C & D \end{pmatrix}$ are presented in terms of $A,B,C,D$ and the generalized Drazin inverses of $A,D$, under the condition that $BD^d=0,~\text{and}~BD^iC=0,~\text{for any nonnegative integer}~ i.$ Using the representation, we give a new additive result of the generalized Drazin inverse for two bounded linear operators $P,Q \in \mathbf{B}(X)$ with $PQ^{d}=0$ and $PQ^{i}P=0$, for any integer $i\geq 1$. As corollaries, several well-known results are generalized.
Banach space, generalized Drazin inverse, operator matrix
• [1] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974.
• [2] S.L. Campbell, Singular Systems of Differential Equations I-II, Pitman, London, San Francisco, 1980.
• [3] S.L. Campbell and C.D. Meyer, Generalized Inverses of Linear Transformations, Dover, New York, 1991.
• [4] N. Castro-González, E. Dopazo and M.F. Matínez-Serrano, On the Drazin inverse of the sum of two operators and its application to operator matrices, J. Math. Anal. Appl. 350 (1), 207-215,2009.
• [5] A.S. Cvetković and G.V. Milovanović, On Drazin inverse of operator matrices, J. Math. Anal. Appl. 375 (1), 331-335, 2011.
• [6] D.S. Cvetković-Ilić, The generalized Drazin inverse with commutativity up to a factor in a Banach algebra, Linear Algebra Appl. 431 (5), 783-791, 2009.
• [7] D.S. Cvetković-Ilić, D.S. Djordjević and Y.M. Wei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl. 418 (1), 53-61, 2006.
• [8] D.S. Cvetković-Ilić, X.J. Liu and Y.M. Wei, Some additive results for the generalized Drazin inverse in a Banach algebra, Electron. J. Linear Algebra 22, 1049- 1058, 2011.
• [9] D.S. Cvetković-Ilić and Y.M. Wei, Representations for the Drazin inverse of bounded operators on Banach space, Electron. J. Linear Algebra 18, 613-627, 2009.
• [10] D.S. Cvetković-Ilić and Y.M.Wei, Algebraic Properties of Generalized Inverses, Series: Developments in Mathematics, 52, Springer, 2017.
• [11] C.Y. Deng, A note on the Drazin inverses with Banachiewicz-Schur forms, Appl. Math. Comput. 213 (1), 230-234, 2009.
• [12] C.Y. Deng, Generalized Drazin inverses of anti-triangular block matrices, J. Math. Anal. Appl. 368 (1), 1-8, 2010.
• [13] C.Y. Deng, D.S. Cvetković-Ilić and Y.M. Wei, Some results on the generalized Drazin inverse of operator matrices, Linear Multilinear Algebra 58 (4), 503-521, 2010.
• [14] C.Y. Deng and Y.M. Wei, A note on the Drazin inverse of an anti-triangular matrix, Linear Algebra Appl. 431 (10), 1910-1922, 2009.
• [15] C.Y. Deng and Y.M. Wei, Representations for the Drazin inverses of 2 × 2 blockoperator matrix with singular schur complement Linear Algebra Appl. 435 (11), 2766- 2783, 2011.
• [16] D.S. Djordjević and P.S. Stanmirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (3), 617-634, 2001.
• [17] D.S. Djordjević and Y.M. Wei, Additive results for the generalized Drazin inverse, J. Austral. Math. Soc. 73 (1), 115-125, 2002.
• [18] E. Dopazo and M. F. Matínez-Serrano, Further results on the representation of the Drazin inverse of a 2×2 block matrix, Linear Algebra Appl. 432 (8), 1896-1904, 2010.
• [19] M.P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly 65 (7), 506-524, 1958.
• [20] L. Guo and X.K. Du, Representations for the Drazin inverses of 2×2 block matrices, Appl. Math. Comput. 217 (6), 2833-2842, 2010.
• [21] R.E. Harte, Spectral projections, Irish Math. Soc. Newsletter 11 (1), 10-15, 1984.
• [22] R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York, 1988.
• [23] R.E. Harte, On quasinilpotents in rings, Pan-Amer. Math. J. 1 (1), 10-16, 1991.
• [24] R.E. Hartwig, and J.M. Shoaf, Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices, Austral J. Math. 24(A), 10-34, 1977.
• [25] R.E. Hartwig, G.R. Wang and Y.M. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (1), 207-217, 2010.
• [26] J.J. Huang, Y.F. Shi and A. Chen, The representation of the Drazin inverses of antitriangular operator matrices based on resolvent expansions, Appl. Math. Comput. 242 (1), 196-201, 2014.
• [27] J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (3), 367-381, 1996.
• [28] J.J. Koliha, The Drazin and Moore-Penrose inverse in $C^*$-algebras, Math. Proc. R. Ir. Acad. 99A (1), 17-27, 1999.
• [29] J.J. Koliha, D.S. Cvetković-Ilić and C. Y. Deng, Generalized Drazin invertibility of combinations of idempotents , Linear Algebra Appl. 437 (9), 2317-2324, 2012.
• [30] J. Ljubisavljević and D.S. Cvetković-Ilić, Additive results for the Drazin inverse of block matrices and applications, J. Comput. Appl. Math. 235 (12), 3683-3690, 2011.
• [31] C.D. Meyer and N.J. Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math. 33 (1), 1-7, 1977.
• [32] G.J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, San Diego, 1990.
• [33] V. Müller, Spectral theory of linear operators and spectral systems in Banach algebras, Operator Theory, Advances and Applications, 139, Birkhäuser Verlag, Basel-Boston- Berlin, 2007.
• [34] H. Yang and X.J. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (5), 1412-1417, 2011.
• [35] G.F. Zhuang, J.L. Chen, D.S. Cvetković-Ilić and Y.M. Wei, Additive property of Drazin invertibility of elements in a ring, Linear Multilinear Algebra 60 (8), 903-910, 2012.
• [36] H.L. Zou, J.L. Chen and D. Mosić, The Drazin invertibility of an anti-triangular matrix over a ring, Stud. Sci. Math. Hung. 54 (4), 489-508, 2017.
• [37] H. L. Zou, D. Mosić and J. L. Chen, The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring, J. Algebra Appl., 18 (11), 2019, doi: 10.1142/S0219498819502128.
Primary Language en Mathematics Mathematics Orcid: 0000-0003-3495-573XAuthor: Li GUO (Primary Author)Institution: Southeast UniversityCountry: China Orcid: 0000-0002-0064-9729Author: Honglin ZOU Institution: Beihua UniversityCountry: China Orcid: 0000-0003-6798-488XAuthor: Jianlong CHEN Institution: Hubei Normal UniversityCountry: China Publication Date : June 2, 2020
 Bibtex @research article { hujms731518, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1134 - 1149}, doi = {10.15672/hujms.731518}, title = {The generalized Drazin inverse of operator matrices}, key = {cite}, author = {Guo, Li and Zou, Honglin and Chen, Jianlong} } APA Guo, L , Zou, H , Chen, J . (2020). The generalized Drazin inverse of operator matrices . Hacettepe Journal of Mathematics and Statistics , 49 (3) , 1134-1149 . DOI: 10.15672/hujms.731518 MLA Guo, L , Zou, H , Chen, J . "The generalized Drazin inverse of operator matrices" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1134-1149 Chicago Guo, L , Zou, H , Chen, J . "The generalized Drazin inverse of operator matrices". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1134-1149 RIS TY - JOUR T1 - The generalized Drazin inverse of operator matrices AU - Li Guo , Honglin Zou , Jianlong Chen Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.731518 DO - 10.15672/hujms.731518 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1134 EP - 1149 VL - 49 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.731518 UR - https://doi.org/10.15672/hujms.731518 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics The generalized Drazin inverse of operator matrices %A Li Guo , Honglin Zou , Jianlong Chen %T The generalized Drazin inverse of operator matrices %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 3 %R doi: 10.15672/hujms.731518 %U 10.15672/hujms.731518 ISNAD Guo, Li , Zou, Honglin , Chen, Jianlong . "The generalized Drazin inverse of operator matrices". Hacettepe Journal of Mathematics and Statistics 49 / 3 (June 2020): 1134-1149 . https://doi.org/10.15672/hujms.731518 AMA Guo L , Zou H , Chen J . The generalized Drazin inverse of operator matrices. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1134-1149. Vancouver Guo L , Zou H , Chen J . The generalized Drazin inverse of operator matrices. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1134-1149.

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