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

TL;DR

This paper introduces a new natural inverse in algebraic structures like semigroups and rings, generalizing existing inverses, and explores its spectral properties in Banach and operator algebras.

Contribution

It defines the natural inverse, unifies various existing inverses, and develops a core decomposition framework applicable to multiple algebraic contexts.

Findings

The natural inverse generalizes Drazin and Koliha-Drazin inverses.

A core decomposition similar to classical decompositions is established.

Spectral analysis in Banach and operator algebras requires local spectral theory.

Abstract

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.