Atlas: The Drazin inverse for closed operators with applications by J. J. Koliha

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18th International Conference on Operator Theory
June 27 - July 1, 2000
University of the West
Timisoara, Romania

Organizers
Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin, Flavius Turcu

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The Drazin inverse for closed operators with applications
by
J. J. Koliha
The University of Melbourne, Australia

The Drazin inverse occurs in a number of applications, such as singular and singularly perturbed differential equations, asymptotic convergence of operator semigroups, Markov chains, cryptography, multibody system dynamics, numerical methods, etc.

Originally defined by Drazin [4] in 1958 in the context of semigroups and rings, it was later generalized to bounded linear operators whose resolvent had a pole, and more generally, any singularity at zero [5]. Applications to semigroups of operators require the Drazin inverse of a closed linear operator A in C(X); a first account was give in 1992 by Nashed and Zhao [11] without proofs-for the case of a resolvent pole. The fully general case is studied in [9].

Let the resolvent of A in C(X) have a (possibly removable) singularity at 0. Then 0 is an isolated spectral point of A with a spectral projection A^\pi in B(X), and we define the Drazin inverse of A by

A^D = (A+A^\pi)^-1(I-A^\pi) ( in B(X)).

In the lecture, we intend to present some of the properties of this generalization of the Drazin inverse, the similarities and differences between the Drazin inverse for bounded and unbounded operators, and discuss some recent advances and applications in Drazin inverse covering the following topics:

singularly perturbed differential equations [8, 10];

perturbation of the Drazin inverse [1, 2];

continuity of the Drazin inverse [3, 7];

study of the Moore-Penrose inverse in C^*-algebras via the Drazin inverse [6].

References

[1] N. Castro González and J. J. Koliha, Perturbation of the Drazin inverse for closed linear operators, Integral Equations Operator Theory 36 (2000), 92-106.

[2] N. Castro González, J. J. Koliha and Yimin Wei, Perturbation of the Drazin inverse for closed operators with equal spectral projections, preprint 2000.

[3] N. Castro González, J. J. Koliha and V. Rakocevi\'c, Continuity and general perturbation of the Drazin inverse for closed linear operators, preprint 2000.

[4] M. P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506-514.

[5] J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367-381.

[6] J. J. Koliha, The Drazin and Moore-Penrose inverse in C^*-algebras, Math. Proc. Roy. Irish Acad. 99 A (1999), 17-27.

[7] J. J. Koliha and V. Rakocevi\'c, Continuity of the Drazin inverse II, Studia Math. 131 (1998), 167-177.

[8] J. J. Koliha and T. D. Tran, Semistable operators and singularly perturbed differential equations, J. Math. Anal. Appl. 231 (1999), 446-458.

[9] J. J. Koliha and T. D. Tran, The Drazin inverse for closed linear operators, J. Operator Theory , to appear.

[10] J. J. Koliha and T. D. Tran, Closed semistable operators and singular differential equations, preprint 1999.

[11] M. Z. Nashed and Y. Zhao, The Drazin inverse for singular evolution equations and partial differential operators, World Sci. Ser. Appl. Anal. 1 (1992), 441-456.

Date received: June 4, 2000