Generalized Inverses in Jordan Algebras

Abstract

The article introduces the concept of generalized inverses in Jordan algebras as an analog to the classical generalized inverses in the absence of the associativity. We define nonassociative generalized inverses based on quadartic operator U a , explore properties of these notion (e.g., existence, characterization, coincidence with the notion of generalized inverse when Jordan algebra is special or associative, compactness of the generalized spectrum introduced here), and prove results analogous to those in associative algebras, including a Jordan version of the generalized resolvent and a conorm.