Weighted m-Generalized Group Inverse in Banach Algebras

We introduce the w-weighted m-generalized group inverse, extending the concept of the W-weighted m-weak group inverse from complex matrices to elements in a Banach algebra. We establish its fundamental properties, representations, and investigate related (weighted) m-generalized core inverses. By employing a limit-based approach, we extend the core theory of generalized inverses to a significantly broader context, establishing a foundational tool for future research in infinite-dimensional settings.