| Title:
|
On the generalized Drazin inverse and generalized resolvent (English) |
| Author:
|
Djordjević, Dragan S. |
| Author:
|
Stanimirović, Predrag S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
617-634 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Drazin inverse |
| Keyword:
|
generalized resolvent |
| Keyword:
|
limit processes |
| Keyword:
|
outer inverses |
| Keyword:
|
operator matrices |
| MSC:
|
46H30 |
| MSC:
|
46L05 |
| MSC:
|
47A05 |
| MSC:
|
47A10 |
| idZBL:
|
Zbl 1079.47501 |
| idMR:
|
MR1851551 |
| . |
| Date available:
|
2009-09-24T10:45:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127673 |
| . |
| Reference:
|
[1] A. Ben-Israel: On matrices of index zero or one.SIAM J. Appl. Math. 17 (1969), 1118–1121. Zbl 0186.34002, MR 0260762, 10.1137/0117102 |
| Reference:
|
[2] A. Ben-Israel and T. N. E. Greville: Generalized Inverses: Theory and Applications.Wiley-Interscience, New York, 1974. MR 0396607 |
| Reference:
|
[3] S. L. Campbell and C. D. Meyer: Generalized Inverses of Linear Transformations.Pitman, New York, 1979. |
| Reference:
|
[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. MR 0431636, 10.1137/0131035 |
| Reference:
|
[5] S. R. Caradus: Generalized Inverses and Operator Theory.Queen’s paper in pure and applied mathematics, Quenn’s University, Kingston, Ontario, 1978. Zbl 0434.47003, MR 0523736 |
| Reference:
|
[6] K.-H. Förster and B. Nagy: Transfer functions and spectral projections.Publ. Math. Debrecen 52 (1998), 367–376. MR 1630828 |
| Reference:
|
[7] C. W. Groetch: Representation of the generalized inverse.J. Math. Anal. Appl. 49 (1975), 154–157. 10.1016/0022-247X(75)90166-3 |
| Reference:
|
[8] W. Guorong: An imbedding method for computing the generalized inverse.J. Comput. Math. 8 (1990), 353–362. MR 1149717 |
| Reference:
|
[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 |
| Reference:
|
[10] R. E. Harte: Spectral projections.Irish Math. Soc. Newsletter 11 (1984), 10–15. Zbl 0556.47001, MR 0762003 |
| Reference:
|
[11] R. E. Harte: Invertibility and Singularity for Bounded Linear Operators.New York, Marcel Dekker, 1988. Zbl 0636.47001, MR 0920812 |
| Reference:
|
[12] R. E. Harte: On quasinilpotents in rings.Panamer. Math. J. 1 (1991), 10–16. Zbl 0761.16009, MR 1088863 |
| Reference:
|
[13] R. E. Harte and M. Mbekhta: On generalized inverses in $C^*$-algebras.Studia Math. 103 (1992), 71–77. MR 1184103, 10.4064/sm-103-1-71-77 |
| Reference:
|
[14] J. Ji: An alternative limit expression of Drazin inverse and its application.Appl. Math. Comput. 61 (1994), 151–156. Zbl 0805.65041, MR 1274303, 10.1016/0096-3003(94)90044-2 |
| Reference:
|
[15] J. J. Koliha: A generalized Drazin inverse.Glasgow Math. J. 38 (1996), 367–381. Zbl 0897.47002, MR 1417366, 10.1017/S0017089500031803 |
| Reference:
|
[16] J. J. Koliha and V. Rakočević: Continuity of the Drazin inverse II.Studia Math. 131 (1998), 167–177. MR 1636348 |
| Reference:
|
[17] I. Marek and K. Žitný: Matrix Analysis for Applied Sciences.Teubner-Texte zur Mathematik, Band 84, Leipzig, 1986. MR 0881384 |
| Reference:
|
[18] C. D. Meyer: Limits and the index of a square matrix.SIAM J. Appl. Math. 26 (1974), 469–478. Zbl 0249.15004, MR 0364284, 10.1137/0126044 |
| Reference:
|
[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. MR 0460351, 10.1137/0133001 |
| Reference:
|
[20] V. Rakočević: Continuity of the Drazin inverse.J. Operator Theory 41 (1999), 55–68. MR 1675243 |
| Reference:
|
[21] N. J. Rose: The Laurent expansion of a generalized resolvent with some applications.SIAM J. Math. Anal. 9 (1978), 751–758. Zbl 0387.40005, MR 0480558, 10.1137/0509054 |
| Reference:
|
[22] U. G. Rothblum: A representation of the Drazin inverse and characterizations of the index.SIAM J. Appl. Math. 31 (1976), 646–648. Zbl 0355.15008, MR 0422303, 10.1137/0131057 |
| Reference:
|
[23] U. G. Rothblum: Resolvent expansion of matrices and applications.Linear Algebra Appl. 38 (1981), 33–49. MR 0636023 |
| Reference:
|
[24] P. S. Stanimirović: Limit representations of generalized inverses and related methods.Appl. Math. Comput. 103 (1999) (to appear). MR 1686357 |
| Reference:
|
[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. |
| Reference:
|
[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 |
| Reference:
|
[27] Y. Wei: A characterization of the Drazin inverse.SIAM J. Matrix Anal. Appl. 17 (1996), 744–747. MR 1410699, 10.1137/S0895479895280697 |
| . |
