Generalized Weighted Group Inverses of Banach Elements

We introduce the generalized weighted group inverse in Banach algebras, a novel concept that unifies the classical group inverse with quasinilpotent elements. We establish its fundamental characterization through an equivalence to a specific polar-like property and elucidate its structural relationship with the generalized weighted core inverse. Employing a novel limit-based approach, we extend the theory of the weak group inverse from matrices and Hilbert space operators to the general setting of Banach algebras, thereby significantly generalizing key existing results.