Perturbation analysis of generalized inverses of linear operators in Banach spaces (Q1887630)

Let \(X_1, X_2\) be Banach spaces and \(T: X_1 \to X_2\) a bounded linear operator with a bounded generalized inverse \(T^+\). The authors study the perturbation problem for the generalized inverse. Specifically, the following stability result is established. Let \(T_0 \in B(X_1,X_2)\) with a bounded generalized inverse \(T_0^+\) and \(T \in B(X_1,X_2)\) with \(\| T_0^+ \| \| T-T_0 \|<1\). Then \(T_0^+ [I_{X_2}+(T-T_0) T_0^+]^{-1}\) is a generalized inverse of \(T\) if and only if \(R(T)\) is closed and \(T |_{R(T_0^+)}: R(T_0^+) \to R(T)\) is an isomorphism. A continuity characterization of the Moore--Penrose inverse in Hilbert space is also given.