Generalizations of weak Drazin inverses for operators

This paper is devoted to the generalizations of weak Drazin inverses from square matrices to the broader framework of generalized Drazin invertible operators. Within this setting, we establish several equivalent characterizations of the newly defined inverses. Furthermore, we introduce the m-minimal rank generalized weak Drazin inverse as a novel extension of the classical definition. By employing suitable block-matrix decompositions, we derive explicit matrix representations for these operators. These representations prove instrumental in deriving new theoretical results and establish a unified framework for the study of weak Drazin inverses in operator theory.