The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces Original Research Article

Let H1,H2 be two Hilbert spaces over the complex field C and let T:H1 → H2 be a bounded linear operator with the generalized inverse T+. Let image be a bounded linear operator with short parallelT+short parallel short parallelδTshort parallel left angle bracket 1. Suppose that dim ker image or image. Then image has the generalized inverse image with image This result gives an analogue of Theorem 3.9 of M.Z. Nashed (“Generalized Inverses and Applications”, Academic Press, New York, 1976) in Hilbert spaces.