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

Abstract

Let H 1, H 2 be two Hilbert spaces over the complex field C and let T: H 1 → H 2 be a bounded linear operator with the generalized inverse T +. Let T ̄ = T + δT be a bounded linear operator with ∥ T +∥ ∥ δT∥ 〈 1. Suppose that dim ker T ̄ = dim ker T 〈 ∞ or R( T ̄ ) ∩ R(T) ⊥ = 0 . Then T ̄ has the generalized inverse T +=[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] −1 with ‖ 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.