On the generalized spectrum of bounded linear operators in Banach spaces

Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent.