On the expression of generalized inverses of perturbed bounded linear operators (Q1407478)

Let \(X\), \(Y\) be Banach spaces (in particular, Hilbert spaces), let \(\Gamma: X\to Y\) be a bounded operator with closed range and let \(T^+\) denote its generalized inverse. The author presents equivalence conditions for the operator \(\widetilde T= T+\delta T\) with \(\|\delta T\|\cdot\| T^+\|< 1\) to have a generalized inverse \[ \widetilde T^+= (I+ T^+\delta T)^{-1} T^+= T^+(I+\delta TT^+)^{-1}. \]