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

AbstractLet 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 T̄ = T + δT be a bounded linear operator with ∥T+∥ ∥δT∥ 〈 1. Suppose that dim ker T̄ = dim ker T 〈 ∞ or R(T̄) ∩ R(T)⊥ = 0. Then T̄ has the generalized inverseT+=[I−(I+T+δT)−1(I−T+T)−(I−T+)(I+T+δT)∗−1]1(1+δTT+)−1T+X[(I+δTT+)TT+)TT+(I+δTT+)−1+(I+δTT+)*ast;−1TT+(I+δTT+)∗−I]−1with‖T+‖⩽‖T+‖1−‖T+‖‖δT‖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