Some new perturbation results for generalized inverses of closed linear operators in Banach spaces (Q437722)

Let \(X\) and \(Y\) be Banach spaces and let \(L(X,Y)\), \(C(X,Y)\), and \(B(X,Y)\) denote the sets of all linear operators (l.o.), of all closed l.o. with a dense domain, and of all bonded l.o. from \(X\) into \(Y\), respectively. Given \(T\in C(X,Y)\) with a generalized inverse \(T^{+}\in B(X,Y)\) and \(\delta T \in L(X,Y)\), which is \(T\)-bounded with the \(T\)-bound less then 1, let the inequality \(\| \delta T T^{+} y\| \leq \lambda _{1} \|y\| + \lambda_{2}\|(I+\delta T T^{+})y\| \;(y\in Y)\) be satisfied with scalars \(\lambda_{1}, \lambda_{2}\in [0,1)\). Then conditions for \(B=T^{+}(I+\delta T T^{+})^{-1} = (I+ T^{+}\delta T)^{-1}T^{+ }: Y \rightarrow X\) to be a generalized inverse of \(\overline{T}=T+\delta T\) are derived in terms of the ranges and the null spaces of the operators involved. The perturbations of the Moore-Penrose inverse of closed l.o., in particular of closed EP l.o., are considered.