Natural generalized invertibility and prescribed idempotents

TL;DR

This paper explores the concept of natural inverse in rings and operators, establishing connections with $(p,q)$-inverses, and characterizes natural invertibility in terms of prescribed range and nullspace, including spectral projections.

Contribution

It introduces new characterizations of natural invertibility and links it with existing $(p,q)$-inverse concepts, especially in operator algebras.

Findings

Characterization of natural invertible operators via prescribed range and nullspace

Connections established between natural inverse and $(p,q)$-inverses

Analysis of spectral projection cases for natural invertibility

Abstract

We study the natural inverse introduced by X. Mary and show some connections with the $(p,q)$-inverses of Djordjevic and Wei, where $p$ and $q$ are prescribed idempotents. We deal first with rings with identity and then specialize to the particular case of the algebra of bounded linear operators. We give a characterization of the set of operators along which an operator is natural invertible in terms of prescribed range and nullspace. Finally, the special case when the prescribed idempotent is the spectral projection is discussed.