Existence Theorems for Generalized Resolvents of Closed Linear Operators in Banach Spaces
Qianglian Huang · Chinese Annals of Mathematics · 2011
The perturbation problem for the generalized inverses and the existence for generalized resolvents of closed linear operators in Banach spaces are studied.The authors first provide some stability characterizations of generalized inverses of closed linear operator under T-bounded perturbation,which improves some well-known results in the case of bounded linear operators,of closed linear operators under bounded perturbation and of that the perturbation does not change the null space and the range of closed linear operators,respectively. Based on these results,some sufficient and necessary conditions for the existence of the generalized resolvents of closed linear operators are obtained.An explicit expression of the generalized resolvent is also given.As applications,the characterizations for the existence of generalized resolvents of closed Predholm operators and closed semi-Predholm operators are also obtained.