We define an extension of weighted G-Drazin inverses of rectangular matrices to operators between two Banach spaces. Some properties of weighted G-Drazin inverses are generalized and some new ones are proved. Using weighted G-Drazin inverses, we introduce and characterize a new weighted pre-order on the set of all bounded linear operators between two Banach spaces. As an application, we present and study the G-Drazin inverse and the G-Drazin partial order for operators on Banach space.

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Mosić, Dijana