Steady Perturbation Theorem Of External Inverse And Its Application

The theory of perturbation and representation of generalized inverses plays a core role in generalized inverses. Many important generalized inverses, such as the Moore-Penrose inverse, the Drazin inverse, the group inverse, are outer inverses. It is well known that outer inverses have lots of applications in many fields such as statistics, optimization, singular operators equation. It is meaningful to investigate systematically the perturbation and representation of outer inverses.Let X and Y be Banach spaces. Let T be a bounded linear invertible operator and T-1be its inverse. It is from Banach Lemma that T-1(I+δTT-1)-1is the inverse of T=T+δT if||δTT-1||<1. It is natural to ask the following problem: Let Tbe an outer inverse of T with||δTT｛2｝||<1, is T｛2｝(I+δTT{2})-1outer inverse of T=T+δT? In1993, M. Nashed and X. Chen pointed out that the answer is positive, i.e., the outer inverse is stable. In2003, J. Ding studied the problem for Moore-Penrose inverse and gave the necessary and sufficient condition for the Moore-Penrose inverse of T=T+δT to have the simplest representation T+(I+δTT+)-1In this paper, we firstly give an example to illustrate that, even in the case of matrics, the {2,3}-,{2,4}-,{2,5}-inverse may not possess stable perturbations in general. Secondly, utilizing some techniques such as the topological decomposition of space, we get the characteristics for I+δTT｛2｝:Y�'Y to be bijective. Thirdly, a new stable perturbation forθ-inverse is defined: Let T∈B(X,Y) have aθ-inverse. An operator T∈5(X, Y) is called to be a stable perturbation of T with respect to Tθ if T has aθ-inverse satisfying R(Tθ)=R(Tθ) and N(Tθ)=N(Tθ). Then we prove that this definition is consistent with the stability of outer inverse in the meaning of M. Nashed and X. Chen, and is equivalent to the stable perturbation of G. Chen and Y. Xue. At last, these results are used to investigate the stable perturbation of Moore-Penrose inverse, Drazin inverse, group inverse and some other outer inverses. Our results improve and extend many known results in [8,11,18,21,25,44]. Theorem Let T∈B(X, Y) have an outer inverse T｛2｝∈B(Y, X). If T=T+δT∈B(X, Y) is a stable perturbation of T with respect to T｛2｝,then I+δTT｛2｝:Y�'Y is bijective and be bijective with δT∈B(X,Y). Then the following statements are equivalent: Theorem Let X and Y be Hilbert spaces. Assume that T∈B(X,Y) has the Moore-Penrose inverse is bijective with δT∈B(X,Y), then the following statements are equivalent: is the Moore-Penrose inverse of T=T+δT; is the Drazin inverse of T=T+δT if and only if the following statements hold:(2) there exists a positive integer...