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

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) over bar = T + delta T be a bounded linear operator with //T+// //delta T// < 1. Suppose that dim ker (T) over bar = dim ker T < infinity or R((T) over bar) boolean AND R(T)(perpendicular to) = 0. Then (T) over bar has the generalized inverse [GRAPHICS] with //(T) over bar(+)// less than or equal to //T+///1 - //T+// //delta T//. 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. (C) 1998 Elsevier Science Inc. All rights reserved.