Left and right generalized Drazin inverses for closed operators and application to singular linear equations

We define and study left and right generalized Drazin inverses for a closed densely defined linear operator in a Hilbert space. These operators are characterized by means of the generalized Kato decomposition and the single-valued extension property. We prove that they are invariant under commuting quasinilpotent perturbations. Finally, we give an application to solve singular linear equations.