The generalized Drazin inverse of operator matrices
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.
___
- [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.