Abstract
The purpose of this paper is to study a new generalized inverse called weighted CMP inverse associated with an operator between two Hilbert spaces, using its Wg-Drazin inverse and its Moore–Penrose inverse. This generalized inverse extends the notion of the weighted CMP inverse for a rectangular matrix. We give some new properties of weighted CMP inverse establishing matrix expression for the weighted CMP inverse of an operator and matrix expression for the Moore–Penrose inverse of this new inverse. Applying these results, we introduce and characterize the CMP inverse for a Hilbert space operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Pitman, London (1979)
Cline, R.E., Greville, T.N.E.: A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 53–62 (1980)
Dajić, A., Koliha, J.J.: The weighted g-Drazin inverse for operators. J. Aust. Math. Soc. 82, 163–181 (2007)
Deng, C.Y., Du, H.K.: Representations of the Moore–Penrose inverse of \(2\times 2\) block operator valued matrices. J. Korean Math. Soc. 46(6), 1139–1150 (2009)
Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)
Drazin, M.P.: Pseudo-inverses in associate rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)
Harte, R.E.: Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker, New York (1988)
Hernández, A., Lattanzi, M., Thome, N.: On some new pre-orders defined by weighted Drazin inverses. Appl. Math. Comput. 282, 108–116 (2016)
Hernández, A., Lattanzi, M., Thome, N.: Weighted binary relations involving the Drazin inverse. Appl. Math. Comput. 253, 215–223 (2015)
Koliha, J.J.: A generalized Drazin inverse. Glasg. Math. J. 38, 367–381 (1996)
Kolundžija, M.Z.: Generalized Sherman–Morrison–Woodbury formula for the generalized Drazin inverse in Banach algebra. Filomat 31(16), 5159–5167 (2017)
Malik, S.B., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226, 575–580 (2014)
Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66(5), 1046–1053 (2018)
Meng, L.S.: The DMP inverse for rectangular matrices. Filomat 31(19), 6015–6019 (2017)
Mosić, D.: Reverse order laws on the conditions of the commutativity up to a factor. Revista de La Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A.Matematicas 111, 685–695 (2017)
Mosić, D.: The CMP inverse for rectangular matrices. Aequ. Math. 92(4), 649–659 (2018)
Mosić, D.: Weighted core–EP inverse of an operator between Hilbert spaces, Linear Multilinear Algebra https://doi.org/10.1080/03081087.2017.1418824
Mosić, D.: Weighted gDMP inverse of operators between Hilbert spaces. Bull. Korean Math. Soc. 55(4), 1263–1271 (2018)
Mosić, D., Djordjević, D.S.: Some additive results for the \(Wg\)-Drazin inverse of Banach space operators. Carpathian J. Math. 32(2), 215–223 (2016)
Mosić, D., Djordjević, D.S.: The gDMP inverse of Hilbert space operators. J. Spect. Theory 8(2), 555–573 (2018)
Moslehian, M.S., Sharifi, K., Forough, M., Chakoshi, M.: Moore–Penrose inverse of Gram operator on Hilbert C*-modules. Stud. Math. 210(2), 189–196 (2012)
Penrose, R.: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)
Rakočević, V., Wei, Y.: A weighted Drazin inverse and applications. Linear Algebra Appl. 350, 25–39 (2002)
Robles, J., Martínez-Serrano, M.F., Dopazo, E.: On the generalized Drazin inverse in Banach algebras in terms of the generalized Schur complement. Appl. Math. Comput. 284, 162–168 (2016)
Srivastava, S., Gupta, D.K., Stanimirović, P.S., Singh, S., Roy, F.: A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element. Sadhana 42(5), 625–630 (2017)
Stanimirović, P.S., Soleymani, F.: A class of numerical algorithms for computing outer inverses. J. Comput. Appl. Math. 263, 236–245 (2014)
Thome, N.: A simultaneous canonical form of a pair of matrices and applications involving the weighted Moore–Penrose inverse. Appl. Math. Lett. 53, 112–118 (2016)
Zhang, X., Sheng, X.: Two methods for computing the Drazin inverse through elementary row operations. Filomat 30(14), 3759–3770 (2016)
Wang, X.Z., Ma, H., Stanimirović, P.S.: Recurrent neural network for computing the W-weighted Drazin inverse. Appl. Math. Comput. 300, 1–20 (2017)
Wang, X., Yu, A., Li, T., Deng, C.: Reverse order laws for the Drazin inverses. J. Math. Anal. Appl. 444(1), 672–689 (2016)
Wang, L., Zhang, S.S., Zhang, X.X., Chen, J.L.: Mixed-type reverse order law for Moore–Penrose inverse of products of three elements in ring with involution. Filomat 28(10), 1997–2008 (2014)
Acknowledgements
The authors are grateful to the referees for the constructive comments and careful reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors are supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, grant no. 174007.
Rights and permissions
About this article
Cite this article
Mosić, D., Kolundžija, M.Z. Weighted CMP inverse of an operator between Hilbert spaces. RACSAM 113, 2155–2173 (2019). https://doi.org/10.1007/s13398-018-0603-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-018-0603-z