Perturbations and expressions for generalized inverses in Banach spaces and Moore–Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore–Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T+(I+δTT+)-1. Utilizing these results, we investigate the expression for the Moore–Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore–Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore–Penrose inverse to have the simplest expression T†(I+δTT†)-1. The results obtained in this paper extend and improve many recent results in this area.