Dagger-Drazin Inverses

TL;DR

This paper introduces the dagger-Drazin inverse, a new generalized inverse concept for arbitrary maps in dagger categories, connecting it to Drazin and Moore-Penrose inverses and providing various examples.

Contribution

It extends the notion of Drazin inverses to arbitrary maps in dagger categories, establishing relationships with Moore-Penrose inverses and providing concrete examples.

Findings

Dagger-Drazin inverses are equivalent to Drazin inverses for positive maps.

A map has a Moore-Penrose inverse if and only if it is a Drazin inverse.

Examples include matrices over fields, bounded linear operators, and partial injections.

Abstract

Drazin inverses are a special kind of generalized inverses that can be defined for endomorphisms in any category. A natural question to ask is whether one can somehow extend the notion of Drazin inverse to arbitrary maps - not simply endomorphisms. It turns out that this is possible and, indeed, natural to do so for dagger categories. This paper, thus, introduces the notion of a dagger-Drazin inverse, which is a new kind of generalized inverse appropriate for arbitrary maps in a dagger category. This inverse is closely related to the Drazin inverse, for having dagger-Drazin inverses is equivalent to asking that positive maps have Drazin inverses. Moreover, dagger-Drazin inverses are also closely related to Moore-Penrose inverses as we observe that a map has a Moore-Penrose inverse if and only if it is a Drazin inverse. Furthermore, we explain how Drazin inverses of opposing pairs correspond precisely to dagger-Drazin inverses in cofree dagger categories. We also give examples of dagger-Drazin inverses for matrices over (involutive) fields, bounded linear operators, and partial injections.