New additive results for the generalized Drazin inverse

AbstractIn this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of PQ=QP, we give explicit representations of the generalized Drazin inverse (P+Q)d in term of P, Pd, Q and Qd. We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases

Some additive results on Drazin inverse

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aa^D(a+b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b)^D. As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.Supported by the National Natural Science Foundation of China (11361009), the Guangxi Provincial Natural Science Foundation of China (2013GXNSFAA019008), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities.Liu, X.; Qin, X.; Benítez López, J. (2015). Some additive results on Drazin inverse. Applied Mathematics - A Journal of Chinese Universities. 30(4):479-490. https://doi.org/10.1007/s11766-015-3333-4S479490304A Ben-Israel, T N E Greville. Generalized Inverses, Theory and Applications, 2nd edition, Springer-Verlag, 2003.S L Campbell, C D Meyer. Generalized Inverses of Linear Transformations, Pitman (Advanced Publishing Program), Boston, MA, 1979.N Castro-González, J J Koliha. Additive perturbation results for the Drazin inverse, Linear Algebra Appl, 2005, 397: 279–297.N Castro-González, E Dopazo, M F Martínez-Serrano. On the Drazin inverse of the sum of two operators and its application to operator matrices, J Math Anal Appl, 2008, 350: 207–215.N Castro-González, M F Martínez-Serrano. Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra, Linear Algebra Appl, 2010, 432: 1885–1895.N Castro-González, J J Koliha. New additive results for the Drazin inverse, Proc Roy Soc Edinburgh Sect A, 2004, 134: 1085–1097.M Catral, D D Olesky, P van den Driessche. Block representations of the Drazin inverse of a bipartite matrix, Electron J Linear Algebra, 2009, 18: 98–107.J L Chen, G F Zhuang, Y Wei. The Drazin inverse of a sum of morphisms, Acta Math Sci Ser A Chin Ed, 2009, 29(3): 538–552.D S Cvetković-Ilić, D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl, 2006, 418, 53–61.D S Cvetković-Ilić. A note on the representation for the Drazin inverse of 2 × 2 block matrices, Linear Algebra Appl, 2008, 429: 242–248.C Deng. The Drazin inverses of sum and difference of idempotents, Linear Algebra Appl, 2009, 430: 1282–1291.C Deng, Y Wei. Characterizations and representations of the Drazin inverse of idempotents, Linear Algebra Appl, 2009, 431: 1526–1538.C Deng, Y Wei. New additive results for the generalized Drazin inverse, J Math Anal Appl, 2010, 370: 313–321.D S Djordjević, P S Stanimirović. On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math J, 2001, 51(126): 617–634.D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse, J Aust Math Soc, 2002, 73: 115–125.D S Djordjević, V Rakočević. Lectures on Generalized inverses, University of Niš, 2008.E Dopazo, M F Martínez-Serrano. Further results on the representation of the Drazin inverse of a 2 × 2 block matrices, Linear Algebra Appl, 2010, 432: 1896–1904.M P Drazin. Pseudo-inverses in associative rings and semiproup, Amer Math Monthly, 1958, 65: 506–514.R E Hartwig, G R Wang, Y Wei. Some additive results on Drazin inverse, Linear Algebra Appl, 2001, 322: 207–217.R E Hartwig, X Li, Y Wei. Representations for the Drazin inverse of a 2×2 block matrix, SIAM J Matrix Anal Appl, 2006, 27: 757–771.Y Liu, C G Cao. Drazin inverse for some partitioned matrices over skew fields, J Nat Sci Heilongjiang Univ, 2004, 24: 112–114.J Ljubisavljević, D S Cvetković-Ilić. Additive results for the Drazin inverse of block matrices and applications, J Comput Appl Math, 2011, 235: 3683–3690.C D Meyer ffixJr, N J Rose. The index and the Drazin inverse of block triangular matrices, SIAM J Appl Math, 1977, 33(1): 1–7.L Wang, H H Zhu, X Zhu, J L Chen. Additive property of Drazin invertibility of elements, arXiv: 1307.1816v1 [math.RA], 2013.H Yang, X Liu. The Drazin inverse of the sum of two matrices and its applications, J Comput Appl Math, 2011, 235: 1412–1417

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

[EN] Based on the conditions ab(2) = 0 and b pi(ab) is an element of A(d), we derive that (ab)(n), (ba)(n), and ab + ba are all generalized Drazin invertible in a Banach algebra A, where n is an element of N and a and b are elements of A. By using these results, some results on the symmetry representations for the generalized Drazin inverse of ab + ba are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b.This work was supported by the National Natural Science Foundation of China (grant number: 11361009, 61772006,11561015), the Special Fund for Science and Technological Bases and Talents of Guangxi (grant number: 2016AD05050, 2018AD19051), the Special Fund for Bagui Scholars of Guangxi (grant number: 2016A17), the High level innovation teams and distinguished scholars in Guangxi Universities (grant number: GUIJIAOREN201642HAO), the Natural Science Foundation of Guangxi (grant number: 2017GXNSFBA198053, 2018JJD110003), and the open fund of Guangxi Key laboratory of hybrid computation and IC design analysis (grant number: HCIC201607).Qin, Y.; Liu, X.; Benítez López, J. (2019). Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra. Symmetry (Basel). 11(1):1-9. https://doi.org/10.3390/sym11010105S19111González, N. C. (2005). Additive perturbation results for the Drazin inverse. Linear Algebra and its Applications, 397, 279-297. doi:10.1016/j.laa.2004.11.001Zhang, X., & Chen, G. (2006). The computation of Drazin inverse and its application in Markov chains. Applied Mathematics and Computation, 183(1), 292-300. doi:10.1016/j.amc.2006.05.076Castro-González, N., Dopazo, E., & Martínez-Serrano, M. F. (2009). On the Drazin inverse of the sum of two operators and its application to operator matrices. Journal of Mathematical Analysis and Applications, 350(1), 207-215. doi:10.1016/j.jmaa.2008.09.035Qiao, S., Wang, X.-Z., & Wei, Y. (2018). Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra and its Applications, 542, 101-117. doi:10.1016/j.laa.2017.03.014Stanimirovic, P. S., Zivkovic, I. S., & Wei, Y. (2015). Recurrent Neural Network for Computing the Drazin Inverse. IEEE Transactions on Neural Networks and Learning Systems, 26(11), 2830-2843. doi:10.1109/tnnls.2015.2397551Koliha, J. J. (1996). A generalized Drazin inverse. Glasgow Mathematical Journal, 38(3), 367-381. doi:10.1017/s0017089500031803Hartwig, R. E., Wang, G., & Wei, Y. (2001). Some additive results on Drazin inverse. Linear Algebra and its Applications, 322(1-3), 207-217. doi:10.1016/s0024-3795(00)00257-3Djordjević, D. S., & Wei, Y. (2002). Additive results for the generalized Drazin inverse. Journal of the Australian Mathematical Society, 73(1), 115-126. doi:10.1017/s1446788700008508Liu, X., Xu, L., & Yu, Y. (2010). The representations of the Drazin inverse of differences of two matrices. Applied Mathematics and Computation, 216(12), 3652-3661. doi:10.1016/j.amc.2010.05.016Yang, H., & Liu, X. (2011). The Drazin inverse of the sum of two matrices and its applications. Journal of Computational and Applied Mathematics, 235(5), 1412-1417. doi:10.1016/j.cam.2010.08.027Harte, R. (1992). On generalized inverses in C*-algebras. Studia Mathematica, 103(1), 71-77. doi:10.4064/sm-103-1-71-77Djordjevic, D. S., & Stanimirovic, P. S. (2001). On the Generalized Drazin Inverse and Generalized Resolvent. Czechoslovak Mathematical Journal, 51(3), 617-634. doi:10.1023/a:1013792207970Cvetković-Ilić, D. S., Djordjević, D. S., & Wei, Y. (2006). Additive results for the generalized Drazin inverse in a Banach algebra. Linear Algebra and its Applications, 418(1), 53-61. doi:10.1016/j.laa.2006.01.015Liu, X., Qin, X., & Benítez, J. (2016). New additive results for the generalized Drazin inverse in a Banach algebra. Filomat, 30(8), 2289-2294. doi:10.2298/fil1608289lMosić, D., Zou, H., & Chen, J. (2017). The generalized Drazin inverse of the sum in a Banach algebra. Annals of Functional Analysis, 8(1), 90-105. doi:10.1215/20088752-3764461González, N. C., & Koliha, J. J. (2004). New additive results for the g-Drazin inverse. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 134(6), 1085-1097. doi:10.1017/s0308210500003632Mosić, D. (2014). A note on Cline’s formula for the generalized Drazin inverse. Linear and Multilinear Algebra, 63(6), 1106-1110. doi:10.1080/03081087.2014.92296

The perturbation of the Drazin inverse and oblique projection

AbstractLet A and E be n × n matrices and B = A+E. Denote the Drazin inverse of A by Ad. We present bounds for ∥Bd∥, ∥BdB∥, ∥Bd − Ad∥∥Ad∥, and ∥BdB − AdA∥∥AdA∥ under the weakest condition core rank B = core rankA. The hard problem due to Campbell and Meyer in [1] is completely solved

Representations for generalized Drazin inverse of operator matrices over a Banach space

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature

Additive Property of Drazin Invertibility of Elements

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and only if $aa^{D}(a-b)bb^{D}$ is Drazin invertible. Next, we give explicit representations of $(a+b)^{D}$, as a function of $a, b, a^{D}$ and $b^{D}$, under the conditions $a^{3}b = ba$ and $b^{3}a = ab$.Comment: 17 page