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Natural Generalized Inverse and Core of an Element in Semigroups, Rings and Banach and Operator Algebras

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Abstract (Original Language): 
Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decompositions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse.
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[1] P. Aiena. Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer
Academic Publishers, Boston, UUS. 2004
[2] A. B. Israel and T.N.E. Greville. Generalized Inverses, Theory and Applications, 2nd
Edition, Springer 2003.
[3] M. P. Dorofeeva. Hereditary and semi-hereditary monoids, Semigroup Forum 4, 301–311.
1972.
[4] M.P. Drazin. Pseudo-Inverses in Associative Rings and Semigroups, American Mathematical
Monthly 65, no. 7, 506–514. 1958
[5] M.P. Drazin. A class of outer generalized inverses, Linear Algebra and its Applications 436,
no. 7, 1909–1923 2012.
[6] J. Fountain. Right PP monoids with central idempotents, Semigroup Forum 13, no. 3,
229–237. 1977.
[7] J. Fountain. Abundant semigroups, Proceedings of the London Mathematical Society 3,
no. 1, 103–129. 1982.
[8] F. Gilfeather. Strong reducibility of operators, Indiana University Mathematics Journal 22,
393–397. 1972.
[9] J.A. Green. On the structure of semigroups, Annals of Mathematics 54, no. 1, 163–172.
1951.
[10] R. Harte. On quasinilpotents in rings, Panamerican Mathematical Journal 1, 10–16.
1991.
[11] D. Kitson and R. Harte. On Browder tuples, Acta Scientiarum Mathematicarum 75, no.
3–4, 665–677. 2009.
REFERENCES 427
[12] R. Harte. On local spectral theory, Recent advances in operator theory and applications
175–183, Operator Theory: Advances and Applications, 187, Birkhäuser, Basel, 2009.
[13] D.A. Herrero and C.L. Jiang. Limits of strongly irreducible operators, and the Riesz decomposition
theorem, Michigan Mathematical Journal 37, no. 2, 283–291. 1990.
[14] G. N. Hile and W. E. Pfaffenberger. Generalized spectral theory in complex Banach algebras,
Canadian Journal of Mathematics 37, no. 6, 1211–1236. 1985.
[15] G. N. Hile and W. E. Pfaffenberger. Idempotents in complex Banach algebras, Canadian
Journal of Mathematics 39 no. 3, 625–630. 1987.
[16] J.J. Koliha. A generalized Drazin inverse, Glasgow Mathematical Journal 38, no. 3, 367–
381. 1996.
[17] J.J. Koliha and P. Patricio. Elements of rings with equal spectral idempotents, Journal of
the Australian Mathematical Society 72, 137–152. 2002.
[18] X. Mary. On generalized inverses and Green’s relations, Linear Algebra and its Applications
434, no. 8, 1836–1844. 2011.
[19] X. Mary and P. Patricio. Generalized invertibilty moduloH in semigroups and rings, Linear
Multilinear Algebra 61, no. 8, 1130–1135. 2013.
[20] M. Gonzalez, M. Mbekhta, and M. Oudghiri. On the isolated points of the surjective spectrum
of a bounded operator, Proceedings of the American Mathematical Society 136, no.
10, 3521–3528. 2008.
[21] M. Mbekhta. Généralisation de la décomposition de Kato aux opérateurs paranormaux et
spectraux, Glasgow Mathematical Journal 29, no. 2, 159–175. 1987.
[22] D.D. Miller and A.H. Clifford. Regular D-Classes in Semigroups, Transactions of the American
Mathematical Society 82, no. 1, 270–280. 1956
[23] H. Radjavi and P. Rosenthal. Invariants Subspaces, Springer-Verlag, Berlin, 1973.

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