The weighted g-Drazin inverse for operators

Abstract

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25-39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53-62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved. © 2007 Australian Mathematical Society.