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Natural generalized inverse and core of an element in semi-groups, rings and Banach and operator algebras. (English) Zbl 1399.47005

Summary: 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.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A11 Local spectral properties of linear operators
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