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The generalized Drazin inverse of operator matrices

Yıl 2020, Cilt: 49 Sayı: 3, 1134 - 1149, 02.06.2020

Li Guo Honglin Zou Jianlong Chen

https://doi.org/10.15672/hujms.731518
Cited By: 3

Öz

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.

Anahtar Kelimeler

Banach space generalized Drazin inverse operator matrix

Kaynakça

  • [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.

Yıl 2020, Cilt: 49 Sayı: 3, 1134 - 1149, 02.06.2020

Li Guo Honglin Zou Jianlong Chen

https://doi.org/10.15672/hujms.731518
Cited By: 3

Öz

Kaynakça

  • [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.
Toplam 37 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Li Guo 0000-0003-3495-573X

Honglin Zou 0000-0002-0064-9729

Jianlong Chen

Yayımlanma Tarihi 2 Haziran 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 49 Sayı: 3

Kaynak Göster

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. 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. Haziran 2020;49(3):1134-1149. doi:10.15672/hujms.731518
Chicago Guo, Li, Honglin Zou, ve Jianlong Chen. “The Generalized Drazin Inverse of Operator Matrices”. Hacettepe Journal of Mathematics and Statistics 49, sy. 3 (Haziran 2020): 1134-49. https://doi.org/10.15672/hujms.731518.
EndNote Guo L, Zou H, Chen J (01 Haziran 2020) The generalized Drazin inverse of operator matrices. Hacettepe Journal of Mathematics and Statistics 49 3 1134–1149.
IEEE L. Guo, H. Zou, ve J. Chen, “The generalized Drazin inverse of operator matrices”, Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 3, ss. 1134–1149, 2020, doi: 10.15672/hujms.731518.
ISNAD Guo, Li vd. “The Generalized Drazin Inverse of Operator Matrices”. Hacettepe Journal of Mathematics and Statistics 49/3 (Haziran 2020), 1134-1149. https://doi.org/10.15672/hujms.731518.
JAMA Guo L, Zou H, Chen J. The generalized Drazin inverse of operator matrices. Hacettepe Journal of Mathematics and Statistics. 2020;49:1134–1149.
MLA Guo, Li vd. “The Generalized Drazin Inverse of Operator Matrices”. Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 3, 2020, ss. 1134-49, doi:10.15672/hujms.731518.
Vancouver Guo L, Zou H, Chen J. The generalized Drazin inverse of operator matrices. Hacettepe Journal of Mathematics and Statistics. 2020;49(3):1134-49.

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