Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras

Xavier Mary (1, 2)

1 MODAL'X - Modélisation aléatoire de Paris X
2 FP2M - Fédération Parisienne de Modélisation Mathématique

THIS IS THE PREPRINT VERSION OF THE ARTICLE. 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.

Mots clés: Koliha-Drazin inverse; generalized inverses

Domaines: Théorie des groupes [math.GR]; Anneaux et algèbres [math.RA]; Algèbres d'opérateurs [math.OA]

Xavier Mary. Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras. European Journal of Pure and Applied Mathematics, 2013, 6 (4), pp.413-427. ⟨hal-00679840⟩