Perturbation analysis for the operator equation \(Tx=b\) in Banach spaces (Q1364832)

The authors provide a perturbation analysis for the generalized inverse \(T^+\) of a bounded linear operator \(T\), with closed range, acting between complex Banach spaces. For perturbations \(\overline{T}\) with \(\delta T= \overline{T}-T\) satisfying \(\| T^+\| \|\delta T\|<1\) and \(R(\overline{T})\cap N(T^+)= \{0\}\) (or \(N(\overline{T})\) and \(N(T)\) finite-dimensional), it is shown that \(\overline{T}^+= (I+ T^+\delta T)^{-1} T^+\) and \(\| \overline{T}^+\|\leq \| T^+\| (1-\| T^+\| \|\delta T\|)\). This generalizes a result of \textit{M. Z. Nashed} [``Generalized inverses and applications'', New York (1976; Zbl 0324.00009)]. The results are applied to a perturbation analysis of the equation \(Tx=b\).