The Stable Perturbation And The Generalized Spectrum Of The Generalized Inverse

The stable perturbation theory of generalized inverse is one of the core content of generalized inverse.In the early 1970s,Professor M.Z.Nashed,a famous expert in generalized inverse research,first discussed the perturbation analysis generalized inverse of linear operators in Banach space.After that,J.P.Ma,G.L.Chen,Y.W.Wang,J.Ding,Y.F.Xue,Y.M.Wei and so on had systemamtically studied the continuity of the Moore-Penrose generalized inverse in Hilbert space and the perturbation stability of the generalized inverse in Banach space.It is well known that the perturbation analysis of generalized inverses has wide applications and plays an important role in many fields,such as computation,optimization,control theory and nonlinear analysis.As to the resolvent,we can investigate the generalized inverse and generalized resolvent.The generalized resolvent has important applications in the rearch of spectral theory and of Fredholm operators.M.A.Shubin showed that there exists a continuous but not analystic generalized inverse function.And he pointed out that it remains open when it exists.This problem has received great concern form many scholars such as C.Badea,M.Mbekhta,S.Christoph.The related results have been widely applied to the generalized spectral theory,Fredholm operator and so on.In this paper,the stable perturbation theory of generalized inverse is used to study the existence of.generalized resolvent and prove that the lacal analyticity of generalized resolution style is equivalent to continuity or locally boundness of generalized inverse function.On this basis,we discussed the relationship of resolvent set,spectral set,the generalized resolvent set and generalized spectrum set and proved generalized resolvent set is open,generalized spectrum set is nonemputy bounded closed set,the spectral radius is equal to the generalized spectral radius and so on.Finally,we discuss the reason why we use generalized inverse rather than the usual Moore-Penrose inverse or group inverse to define generalized resolvent.