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On the generalized resolvent of linear pencils in Banach spaces

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $λ\rightarrow T-λS$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.