×

The generalized Drazin inverse of operator matrices. (English) Zbl 1488.15005

Summary: 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 \(BD^d=0\), and \(BD^iC=0\), 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 \(PQ^iP=0\), for any integer \(i\geq 1\). As corollaries, several well-known results are generalized.

MSC:

15A09 Theory of matrix inversion and generalized inverses
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [1] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974. · Zbl 0305.15001
[2] [2] S.L. Campbell, Singular Systems of Differential Equations I-II, Pitman, London, San Francisco, 1980. · Zbl 0419.34007
[3] [3] S.L. Campbell and C.D. Meyer, Generalized Inverses of Linear Transformations, Dover, New York, 1991. · Zbl 0732.15003
[4] [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. · Zbl 1157.47001
[5] [5] A.S. Cvetković and G.V. Milovanović, On Drazin inverse of operator matrices, J. Math. Anal. Appl. 375 (1), 331-335, 2011. · Zbl 1207.47002
[6] [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. · Zbl 1214.47005
[7] [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. · Zbl 1104.47040
[8] [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. · Zbl 1252.15006
[9] [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. · Zbl 1187.47004
[10] [10] D.S. Cvetković-Ilić and Y.M.Wei, Algebraic Properties of Generalized Inverses, Series: Developments in Mathematics, 52, Springer, 2017. · Zbl 1380.15003
[11] [11] C.Y. Deng, A note on the Drazin inverses with Banachiewicz-Schur forms, Appl. Math. Comput. 213 (1), 230-234, 2009. · Zbl 1182.47001
[12] [12] C.Y. Deng, Generalized Drazin inverses of anti-triangular block matrices, J. Math. Anal. Appl. 368 (1), 1-8, 2010. · Zbl 1191.15004
[13] [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. · Zbl 1196.15009
[14] [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. · Zbl 1177.15003
[15] [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. · Zbl 1225.15006
[16] [16] D.S. Djordjević and P.S. Stanmirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (3), 617-634, 2001. · Zbl 1079.47501
[17] [17] D.S. Djordjević and Y.M. Wei, Additive results for the generalized Drazin inverse, J. Austral. Math. Soc. 73 (1), 115-125, 2002. · Zbl 1020.47001
[18] [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. · Zbl 1187.15004
[19] [19] M.P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly 65 (7), 506-524, 1958. · Zbl 0083.02901
[20] [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. · Zbl 1205.15013
[21] [21] R.E. Harte, Spectral projections, Irish Math. Soc. Newsletter 11 (1), 10-15, 1984. · Zbl 0556.47001
[22] [22] R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York, 1988. · Zbl 0636.47001
[23] [23] R.E. Harte, On quasinilpotents in rings, Pan-Amer. Math. J. 1 (1), 10-16, 1991. · Zbl 0761.16009
[24] [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. · Zbl 0372.15003
[25] [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. · Zbl 0967.15003
[26] [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. · Zbl 1334.15011
[27] [27] J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (3), 367-381, 1996. · Zbl 0897.47002
[28] [28] J.J. Koliha, The Drazin and Moore-Penrose inverse in \(C^*\)-algebras, Math. Proc. R. Ir. Acad. 99A (1), 17-27, 1999. · Zbl 0943.46031
[29] [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. · Zbl 1259.46038
[30] [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. · Zbl 1223.15007
[31] [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. · Zbl 0355.15009
[32] [32] G.J. Murphy, \(C^*\)-Algebras and Operator Theory, Academic Press, San Diego, 1990. · Zbl 0714.46041
[33] [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. · Zbl 1208.47001
[34] [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. · Zbl 1204.15017
[35] [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. · Zbl 1261.15009
[36] [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. · Zbl 1399.15031
[37] [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. · Zbl 1421.15002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.