Abstract
We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Wei and Wang, Castro and Koliha, Rakocevic and Wei, Castro and Koliha and Wei.
Highlights
Introduction and PreliminariesLet X be a Banach spaces
3–5, we know that T is generalized Drazin invertible if and only if 0 is not an accumulation of σ T
We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space
Summary
As an important extension of the conventional Drazin inverse, the generalized Drazin inverse in Banach algebra was introduced firstly by Koliha 1. Later, this notion was extended to closed linear operators by Koliha and Tran 2. An operator T ∈ C X that possesses a generalized Drazin inverse is said to be generalized Drazin invertible, and its generalized Drazin inverse is denoted by T d. We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Castro Gonzalez and Koliha 3 , Castro Gonzalez et al 4 , Wei and Wang 6 , Rakocevicand Wei 7
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