Left and right resolvents and new characterizations of left and right generalized Drazin invertible operators

Abstract

Left and right resolvents of left and right generalized Drazin invertible operators are introduced in this paper. The construction of left and
right resolvents allows us to find, in terms of the coefficients of Laurent series,
new representation results for left and right generalized Drazin inverses and
the associated spectral projections. Fundamental characterizations of left and
right generalized Drazin invertible operators are also obtained, using essentially the range, the quasi-nilpotent part and the analytic core.