Banach Spaces Closed Operator Generalized Drazin Inverse Perturbation Representation Theorem

The theory of generalized inverses has become an important branch in modern mathematics and has substantial content, such as generalized inverses of matrix, the generalized inverses of linear transformations in linear space, the Moore-Penrose inverses of linear operator in Hilbert space, the generalized inverses of linear operator in Banach space. Generalized inverse plays an important role in many areas. Generalized inverse has become an indispensable tool, especially in the least squares problems, ill-posed problems, regression, estimation of distribution and Markov chains.One core of the theory of generalized inverses is its perturbation theory which studies the problem whether the operator, after a small perturbation, still has generalized inverse and whether the generalized inverse convergences to the original one. This kind of problem has important significance in many areas.As an important kind of generalized inverses, Drazin inverse was introduced by Drazin in1958. Later, Koliha extended it to generalized Drazin inverse. Properties for Drazin inverses and group inverses, such as property of spectrum, make them play improtent roles in matrix computation and matrix application.Using the methods of perturbation analysis for Drazin inverses of bounded linear operators and closed linear operators in Banach spaces, we firstly investigate the stable perturbation of the generalized Drazin inverses of closed linear further. Secondly, as corollarys, we recover the perturbation theorems of Koliha, Castro-Gonzdlez, Wei and Wang. Finally, as applications, we use our perturbation theorems to investigate the perturbation problem of group inverse and EP operator in the case of closed linear operator. Theorem Let T∈C(X) have the generalized Drazin inverse Td∈B(X), and let AT△L(X) satisfy D(T)(?) D(AT) and T=T+△T C(X). Then the following statements are equivalent:(1) T is generalized Drazin invertible with Tπ=Tπ;(2) T is generalized Drazin invertible with N(Td)=N(Td) and R(Td)=R(Td); (3)I+△TTd:X→X is bijective and B=Td(I+△TTd)-1=(I+Td△T)-1Td is generalized Drazin inverse of T；(4)T is generalized Drazin invertible with Td-Td=Td△TTd=Td△TTd；(5)T is generalized Drazin invertible with Td-Td=Td△TTd.Theorem Let T∈C(X) have the generalized Drazin inverse Td∈B(X),and let△T∈L(X) satisfy D(T)(?)D(△T)and T=T+△T∈C(X).Assume that I+△TTd:X→X is bijective,then the following statements are equivalent:(1) B=Td(I+△TTd)-1=(I+Td△T)-1Td is the generalized Drazin inverseofT；(2)TTd-TTdTTd,TdT=TdTTdT and limn→+∞‖T(I-TdT)‖1/n=0(3)TTdT=TTdT and limn→+∞‖T(I-TdT)‖1/n=0(4)TTdT=TTdT and TTπ is quasinilpotent;(5)TTd=TTdTTd,TdT=TdTTdT and TTπ is quasinilpotent. In this case,Tπ=Tπ and‖Td-Td‖≤‖Td‖·‖△TTd‖·‖I+△TTd)-1‖.