# Article

Full entry | PDF   (0.3 MB)
Keywords:
Drazin inverse; generalized resolvent; limit processes; outer inverses; operator matrices
Summary:
We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in $C^*$-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, $2\times 2$ operator matrices are considered. As corollaries, we get some well-known results.
References:
[1] A. Ben-Israel: On matrices of index zero or one. SIAM J. Appl. Math. 17 (1969), 1118–1121. DOI 10.1137/0117102 | MR 0260762 | Zbl 0186.34002
[2] A. Ben-Israel and T. N. E. Greville: Generalized Inverses: Theory and Applications. Wiley-Interscience, New York, 1974. MR 0396607
[3] S. L. Campbell and C. D. Meyer: Generalized Inverses of Linear Transformations. Pitman, New York, 1979.
[4] S. L. Campbell, C. D. Meyer and N. J. Rose: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31 (1976), 411–425. DOI 10.1137/0131035 | MR 0431636
[5] S. R. Caradus: Generalized Inverses and Operator Theory. Queen’s paper in pure and applied mathematics, Quenn’s University, Kingston, Ontario, 1978. MR 0523736 | Zbl 0434.47003
[6] K.-H. Förster and B. Nagy: Transfer functions and spectral projections. Publ. Math. Debrecen 52 (1998), 367–376. MR 1630828
[7] C. W. Groetch: Representation of the generalized inverse. J. Math. Anal. Appl. 49 (1975), 154–157. DOI 10.1016/0022-247X(75)90166-3
[8] W. Guorong: An imbedding method for computing the generalized inverse. J. Comput. Math. 8 (1990), 353–362. MR 1149717
[9] W. Guorong and Y. Wei: Limiting expression for generalized inverse $A_{T,S}^{(2)}$ and its corresponding projectors. Numerical Mathematics (A Journal of Chinese Universities) 4 (1995), 25–30. MR 1358556
[10] R. E. Harte: Spectral projections. Irish Math. Soc. Newsletter 11 (1984), 10–15. MR 0762003 | Zbl 0556.47001
[11] R. E. Harte: Invertibility and Singularity for Bounded Linear Operators. New York, Marcel Dekker, 1988. MR 0920812 | Zbl 0636.47001
[12] R. E. Harte: On quasinilpotents in rings. Panamer. Math. J. 1 (1991), 10–16. MR 1088863 | Zbl 0761.16009
[13] R. E. Harte and M. Mbekhta: On generalized inverses in $C^*$-algebras. Studia Math. 103 (1992), 71–77. MR 1184103
[14] J. Ji: An alternative limit expression of Drazin inverse and its application. Appl. Math. Comput. 61 (1994), 151–156. DOI 10.1016/0096-3003(94)90044-2 | MR 1274303 | Zbl 0805.65041
[15] J. J. Koliha: A generalized Drazin inverse. Glasgow Math. J. 38 (1996), 367–381. DOI 10.1017/S0017089500031803 | MR 1417366 | Zbl 0897.47002
[16] J. J. Koliha and V. Rakočević: Continuity of the Drazin inverse II. Studia Math. 131 (1998), 167–177. MR 1636348
[17] I. Marek and K. Žitný: Matrix Analysis for Applied Sciences. Teubner-Texte zur Mathematik, Band 84, Leipzig, 1986. MR 0881384
[18] C. D. Meyer: Limits and the index of a square matrix. SIAM J. Appl. Math. 26 (1974), 469–478. DOI 10.1137/0126044 | MR 0364284 | Zbl 0249.15004
[19] C. D. Meyer, Jr. and N. J. Rose: The index and the Drazin inverse of block triangular matrices. SIAM J. App. Math. 33 (1977), 1–7. DOI 10.1137/0133001 | MR 0460351
[20] V. Rakočević: Continuity of the Drazin inverse. J. Operator Theory 41 (1999), 55–68. MR 1675243
[21] N. J. Rose: The Laurent expansion of a generalized resolvent with some applications. SIAM J. Math. Anal. 9 (1978), 751–758. DOI 10.1137/0509054 | MR 0480558 | Zbl 0387.40005
[22] U. G. Rothblum: A representation of the Drazin inverse and characterizations of the index. SIAM J. Appl. Math. 31 (1976), 646–648. DOI 10.1137/0131057 | MR 0422303 | Zbl 0355.15008
[23] U. G. Rothblum: Resolvent expansion of matrices and applications. Linear Algebra Appl. 38 (1981), 33–49. MR 0636023
[24] P. S. Stanimirović: Limit representations of generalized inverses and related methods. Appl. Math. Comput. 103 (1999) (to appear). MR 1686357
[25] Y. Wei: A survey on the generalized inverse $A_{T,S}^{(2)}$. Actas/Proceedings, Meetings on Matrix Analysis and Applications, Sevilla, Spain, (EAMA), Sep. 10–12,, 1997, pp. 421–428.
[26] Y. Wei: A characterization and representation of the generalized inverse $A_{T,S}^{(2)}$ and its applications. Linear Algebra Appl. 280 (1998), 87–96. MR 1645022
[27] Y. Wei: A characterization of the Drazin inverse. SIAM J. Matrix Anal. Appl. 17 (1996), 744–747. DOI 10.1137/S0895479895280697 | MR 1410699

Partner of