Outer generalized inverses with prescribed ideals

Abstract

The outer inverse of a bounded linear operator on a Banach space is unique if we fix its range and nullspace. In this note we review several approaches to the uniqueness of the outer inverse in rings and explore the underlying idea of prescribing certain ideals in a way that mimics the operators case. Starting with Djordjevic and Wei -inverses, we show some connections with Mary’s inverse along an element and give a characterization of Drazin’s annihilator -inverse.

Keywords:

• Outer generalized inverse
• idempotent
• principal ideal
• annihilator

AMS Subject Classification:

• 15A09
• 47A05

Acknowledgements

The author is indebted to an anonymous referee for valuable suggestions that contributed to shape the final form of this article. This research was partially supported by CONACYT, Mexico, under grant CB-2011-01. The warm hospitality of Stephen B. Sontz during a postdoctoral fellowship at CIMAT is gratefully acknowledged (Project: 160208 ‘Teoría Fredholm en álgebras de Banach mediante inversas generalizadas’).