Dagger-Drazin Inverses

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.