The Generalized <em>w</em>-Core Inverse in Banach Algebras with Involution

In this paper, we present the concept of the generalized w-core inverse within the context of a Banach *-algebra. We delineate the characteristics of this new generalized inverse by using the generalized weighted core decomposition and elucidate its representations via the weighted g-Drazin inverse. Additionally, we delve into an examination of the generalized w-core orders. This work broadens the scope of the weighted core inverse and pseudo core inverse, previously defined for complex matrices and linear bounded operators, to a more general setting.