Generalized Core Inverse with Respect to a Pair of Elements

In this paper, we establish the coincidence conditions for the generalized core-inverse and (b,c)-inverse of a Banach element. We introduce the generalized core-(b,c)-inverse, a unifying extension characterized via quasinilpotents. A novel limit-based approach is developed to investigate this generalized inverse, yielding polar-like properties and image-based representations that extend the core-(b,c) inverse framework.