Dagger-Drazin Inverses
Robin Cockett (University of Calgary), Jean-Simon Pacaud Lemay (Macquarie University), Priyaa Varshinee Srinivasan (Tallinn University of Technology)

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…
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Taxonomy
TopicsBiotin and Related Studies · Melanoma and MAPK Pathways
